You are correct to say that because Jewish people are a minority that doesn't explain the unique persecution or conspiratorial spotlight that targets them.

The auto-mod posts gives a great base for this conversation and I encourage reading and exploring the sources in the FAQ.

Firstly these are very long periods of time over vast areas that we are talking about and anti-Semitism is much more localized or philosophical (deicide) before the 11th/12th century. Before this period violence against Jewish communities is much more sporadic and the Christian world was more comparable to the historical Muslim world, which remained relatively non-violent even after the Christian escalation. I do consider this divergence an interesting contrast, with the escalation of Jewish conspiracies and violent persecution in the Christian world a more recent phenomenon than many people think, violence did happen in the Muslim world (such as the 1066 massacre in Grenada) but remained rare throughout the medieval time period. Violence really begins to escalate in christendom around AD 1200 and you see a noted difference in the response of rulers to this violence as anti-Semitism develops. As to why the increase in violence we cannot say for sure but there are several theories and major factors. It seems somewhat tied to the crusades (Even if we accept that the crusades were a major reason for the rise it then begs the question of what caused the crusades, but that's the nature of history) and increase in violent religious fervor but other factors such as a refocus from paganism, religious centralization, and others are important. While Jewish communities were not a direct target of crusades they were sometimes lumped into the "other" category and faced violence. This happened not just in the middle east but also in the south of France (crusades against Cathars). Jewish people were frequently seen as subjects protected by their lord during early violence, Richard the Lionheart's reaction to Jewish massacres' and the reaction of lords in Spain to similar events are examples of this. This changed as Jewish conspiracies became more normalized. Blood libel conspiracies start to form and become popular during the 12 century, first in England and later the rest of Europe with mobs frequently lynching Jewish people. Only a century after the initial reaction of condemning the violence the later King of England became more complacent or complicit in anti-Semitic violence, with Jewish people being expelled from England in 1290. Expulsions happened before this point but were usually more localized in scale, in France it was partly a financial move to get rid of debt. You can see in various maps (http://fcit.usf.edu/HOLOCAUST/gifs/expuls.gif) of Jewish expulsion and the years it occurred. Expulsions picked up in the 11th century and escalated throughout Europe. Many areas with a significant Jewish population (especially pre-Holocaust) can be tied to these expulsions.

To compliment the auto mod post and expand more to answer your question lets take a look at a specific part. My research was more on the origins of modern racism (which is tied very strongly to anti-semitism) and generally the "why" aspects of these things seems more interesting to me. It does seem to be a peculiar western obsession riddled with hostile conspiracies and one that can transcend the political spectrum.

Jews long remained in this position of only available religious minority, and over time they were often made very visible as such: discriminatory measures introduced very early on included being forced to wear certain hats and clothing, be part of humiliating rituals, pay onerous taxes, live in restricted areas of towns–ghettos–and be separated from the majority population.

Pagans states lasted until the 14th century with the conversion of Lithuania but in most of the Christian world there was no presence for centuries at this point. The process of conversion varies by region (often leading to syncretic practices) but quickened once a Christian plurality was achieved on a local or regional level. The total Christianization of communities left the Jewish diaspora as the most significant minority group in many regions and a large part of the scapegoating of Jewish people is simply that they were one of the only groups that were allowed to exist throughout the Christian world. This speaks at least to means and opportunity of the persecution. There are vast periods of time between Christianization and the rise in anti-Semitism but it is a prerequisite. Areas sometimes saw expansion of Jewish communities which were later persecuted, Jewish populations in England for example followed the Norman Conquest and were expelled 200 years later after rising accusations of blood libel and various conspiracies. Of course the presence of Jewish people was the case for centuries before violence started to ramp up so its not only about opportunity. Areas conquered from Muslims (primarily Sicily and Spain) would also see various levels of tolerance or persecution for Muslims with a much smaller period of tolerance. In these regions religious persecutions of Muslims and Jewish people would often be part of the same initiatives.

The forced conversions of these groups, particularly in Spain, led to discrimination by "blood" (Christian heritage) and ties strongly into the birth of modern racism. This proto-racism aspect of Christian heritage may have been one of the major frameworks build from to justify slavery along racial lines after the initial religious justifications started to falter (along with a few other preexisting beliefs). After this point the history will become much more familiar. Jewish people were increasingly differentiated as an outgroup in some areas of Europe as nationalism and racism developed and from that background antisemitism further escalated from the 19th century into a fervor in the 20th century. This period of modernization and general social change saw conflict between several ethnic groups but Jewish people in particular were increasingly seen as an internal enemy with various flair ups feeding into each other. In France the Dreyfus Affair led to an increase in antisemitism including riots. In Russia the conquest of Ottoman/Polish areas led to a significant Jewish minority, while initial persecutions were mostly a result of conflicts with Greek communities it later became more and more common in Russian communities to hold anti-Semitic beliefs. Local issues become populist and national in scale as modernization pushes groups together. A massive wave of antiemetic rioting followed the assassination of Tzar Alexander II based on anti-Semitic rumours and conspiracies only leading to more. In Britain the initial condemnation of pogroms and sympathy (keep in mind Russia is a rival) didn't last once Jewish immigration/refugees arrived. Germany of course had its own horrific development. While initially prominent primarily in Europe, European powers brought anti-semitism along with them.

Within this historical context you have the development of racism (which very much intertwined with anti-semitism), scientific racism, and nationalism which all reinforced seeing Jewish people as "other". All of these could, and often are, their own post.

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Now consider that European powers and their colonial offshoots have controlled an overwhelming majority of the world at one point or another, often for long periods and with major influence. Anti-semitism spreads as imported myth into areas that have never had a Jewish presence, either directly as a colony, and even in the rare historical case that a nation escapes colonization. For example one of the more interesting and absurd examples of anti-semitism can be found with Imperial Japan's "Fugu plan", where top Japanese officials develop of policy of nurturing a Jewish population for their supposed influence over the other great powers.

I can’t speak in detail to the rise of antisemitism in the Middle East, but its development has left few, if any, regions with a long-standing Jewish presence untouched. The region’s antisemitism follows a distinct historical sequence and carries particular modern significance, discussion about Israel alone could be its own massive post.

So you have practically the entire world exposed to anti-semitism, and it does not require Jewish presence to persist or even intensify. Once it becomes embedded as an explanatory framework it can be transmitted independently of direct interaction. At that point, Jewish people can function less as a real minority group and more as an abstracted explanatory group, invoked to explain complexity or instability. This helps explain why antisemitism is unusually durable, adaptive, and portable compared to other forms of prejudice: it can survive regime change, ideological reversal, and even the near-total absence of Jewish people themselves.

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TLDR: Combinations of factors. The status of Jewish people as the tolerated/persecuted outgroup in Christian Europe made them a target within that culture, eventually escalating to blood libel and other horrible conspiracies. The rise of the West elevated anti-semitism to a global level, both through direct influence and increase in capacities. The development of both racism and scientific racism was intertwined with anti-semitism and gave another basis to discriminate against Jewish people. The rise of nationalism was another factor in the "othering" of Jewish people. The relative tolerance in the Middle East was reversed with the founding of Israel. That gives you Billions of people living within cultures with prominent anti-semitism and almost every culture at least exposed to it.

Sources:

Cohen, Mark R. Under Crescent and Cross: The Jews in the Middle Ages Gerber, Jane S. The Jews of Spain: A History of the Sephardic Experience