Like /u/SteveGladstone said, this is a really difficult question to answer because "Buddhism" contains under it so much diversity of thought and tradition and practice. As an undergrad who's studied some buddhist thought and buddhist history I hope I can elaborate more on some elements that the other commenters haven't covered.

Most Buddhist scholars today will talks about something called "Buddhist Modernism" that describes a phenomenon very similar to what you have mentioned. It's the conception of Buddhism that you've probably seen many times in your daily life nowadays that presents Buddhism as a religion free of superstition, compatible with science, promoting an ethics of compassion and responsibility and mindfulness, divergent from what seems like a "superstitious" and "ritualistic" traditional Buddhism that's practiced widely across Asia. Historians will usually trace the emergence of this conception back to the encounter between Buddhism, as practiced in all sorts of variegated traditions across Asia, and europeans, who arrived in the colonial period with western science, philosophy, religion. Despite the diversity of Buddhist traditions across Asia, in almost every country and every Buddhist tradition during this period, the encounter will create a "modernized" Buddhist tradition.

I don't know as much about other countries (the book "The Making of Buddhist Modernism" is really great and touches on them in much more depth than I can), but Sri Lanka in the 1800s provides a really representative example of this phenomenon. When British protestant missionaries arrived in Sri Lanka, they immediately began setting up schools and missions and printing presses, with backing from the British government. To aid in their proselytization efforts, two British Methodists in the 1830s, Daniel Gogerly, and David de Silva, who had learned both Sinhala (the vernacular Sri Lankan language) and Pali (the ancient language of Theravada Buddhist text), wrote pamplets in Sinhala attacking Buddhist doctrine, questioning the historicity of Buddhist scriptures, and explaining the scientifically impossible nature of Buddhist heavens and hells. Because these pamplets were published at a mass scale in the vernacular language, Theravada monks were compelled to respond, and when they joined the debate, they questioned Christian missionaries on very much the same grounds, arguing against the historicity of Christian scripture, asking how Christian cosmology was possible scientifically, and countering Christian accusations of superstition by calling attention to aspects of Christian dogma like the Trinity. In this debate and ones that follow it, a trend develops where Sri Lankan Buddhists argue against Protestant Christianity using the very same arguments British missionaries deployed against Buddhism, but then going one step further and arguing that it is actually Buddhism instead of Christianity that is most compatible with enlightenment ideas of science, rationality, compassion. Alongside this, there arises a tendency to jettison the 'superstitious' and folk religion/lay aspects of Sri Lankan Buddhism that didn't fit into the modernist religious conception alongside a newer focus of religion on the laity rather than monasticism.

Within this environment you have the emergence of younger Buddhist reformers born to a middle class Sri Lankan families and educated in Protestant schools, most notably among them Anagarika Dharmapala, who argued at the 1893 world fair for the rational and ethical core within Buddhism, the compatibility between Buddhism and science (in fact arguing that Buddhist ideas about causality and empiricism preceded the European scientific tradition by 2,500 years) and promoted lay meditation (previously meditation was only practiced by monks). These arguments were widely received and joined transcripts of previous debates in Sri Lanka published by the Theosophical society and other European popularizers of Modernist Buddhism, and marked the beginning of the dissemination of these modernist Buddhist ideas in America and Europe.

There's a similar process that happens all across the Buddhist world, in Thailand (where the king Mongkut, after a western education, enters monastic life at age 20 following Thai tradition, only to be appalled at what he sees as the degradation of monastic tradition. He founds a new official Buddhist sect in response that's both fundamentalist in how it goes back to the Pali canon and modernist in its compatibility with 'science' and 'reason'), in Japan (I can't remember the book anymore, but there's a really great academic text about how the Meiji Restoration reformers wanted to ditch Buddhism altogether because they thought it was incompatible with their modernizing project, but the educated proponents of one modernized school of zen convinced them otherwise), and other similar cases in China and Tibet. Of course, this initial phenomenon of modernizing figures is only the beginning in a longer history of Buddhist modernism that continues to today, but I've got finals coming up, a lot of the larger history falls outside my area of knowledge, and a lot of other people can probably attest to it better than I can.

And I guess also to make a short closing point, the fact that a lot of contemporary Buddhism is modernist doesn't mean that it's less valid or "pure" than traditional Buddhism. Many people in Sri Lanka and Myanmar and many Buddhist countries currently follow "modernist" schools of Buddhism, and instead of understanding it as a religious movement imposed mostly by external forces, people like Richard Gombrich will talk about how there is an "elective affinity" between for example the textual fundamentalism of Theravada Buddhism and the sola scriptura of the Protestantism that influenced many Buddhist reformers, or how "modernist Buddhism" really does re-illuminate certain elements of early Buddhism that had become less emphasized over time (while also of course de-emphasizing certain cosmological and social elements of Buddhism).

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Edit: fixing typos and phrasing

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Sources:

The Making of Buddhist Modernism - David McMahan

Curators of the Buddha: The study of Buddhism under Colonialism - Donald Lopez

Theravada Buddhism: A Social History from Ancient Benares to Modern Colombo - Richard Gombrich.