Antisemitism has a looooooong history and it's a complicated one at that. So any answer you get here is going to be partial at best. For the sake of ease, let's break it into three brief parts: theological, medieval and early modern socio-cultural, and modern racial/ethnic. This is all a radical over simplification, but it gets to the gist.

First, there's the theological issue. This, properly speaking, isn't necessarily antisemitic and can be framed in legitimate theological controversy and be anti-Judaism, but in broad strokes, part of the history of antisemitism is antagonism towards Jews not embracing the new traditions of Christianity and, later, Islam. In Christianity, this theological antipathy dovetails with part of the medieval issue, which was that Jews were blamed for the crucifixion of Jesus.

Second, in medieval Europe, Jews, like many diaspora communities throughout history, often served as money changers and money lenders and merchants. They operated in a middle space within various parts of European medieval societies and, above and beyond the antipathy which arose from theological/communal issues, they could often become targets or scapegoats when times were tough, money was tight, and debts were hard to pay off. Furthermore, in times of peculiar upheaval, like in the plague years or the Reformation, their outsider status within the communal world of medieval and early-modern Europe made them targets. The languages of this outside status persists and morphs with ...

Third, the modern racial issue. As a perpetual outsider minority, the arrival of ideas of ethno-nationalism helped create an identity for the Jews as both a religion and a nation. As Germanic ideas of kultur morphed (through the prism of Darwinism and social Darwinism) into the concept of biological race, the Jews became identified again as something peculiarly "other" and foreign to Western/European societies. (This is an enormous over simplification and I can o on at length about the 18th century cultural, philosophical, and religious roots of the shifts in the 19th C).

All of these traditions persist to today and have migrated from the boundaries of Europe to the Americas, Africa, and Asia, particularly (but by no means exclusively) the Middle East where Zionism and its complex history and politics made relatively uneducated and broadly politically oppressed peoples receptive to simplistic, reductive, and prejudicial views of the Jews and Judaism. (I hasten to add that, I neither intend to defend Zionism nor justify antisemitism).

So, in broad strokes, its all very complicated and goes back many centuries and it sucks.