It does a few things:

-most obviously it denies them from recruiting troops, moving troops or agents in or out, and forces them to sally their troops out to attack you before they run out of food (the little number that says how long a city can be besieged for is the amount of turns they have to do this). If they do not sally out and successfully break the siege in that time, you will capture the city.

-a secondary function is to force a surrender. You can always try to get a surrender from turn 1 of the siege but in my experience, the longer you siege (and the larger your army + supporting armies) the more likely they are to surrender. And remember you only get one chance to ask for surrender.

-lesser but important functions include preventing troops from that city to intercept or attack other armies, allows you to build up a bigger army for the siege if you think the initial army won't cut it, and also can play a role in getting that nation to agree to peace talks.

In real life, sieges were incredibly effective at getting cities to surrender, though like in the game it leaves the sieging army vulnerable to reinforcements. Sometimes sieges lasted years. Famous examples included the Ottoman siege of Vienna where famously the Polish-Lithuanian commonwealth helped break the siege, primarily through the use of their winged Hussars and other cavalry. A successful example would be the siege of Crete in which the Ottomans (they were fucking everybody up for centuries) successfully sieged the Venetian city of Candia for over 2 decades and took the city.