It began really with one of the four main books in the new testament which says the jews encouraged the Romans to kill christ. Interestingly, only one of the books tell this story although all four essentially are telling the same story. (The fact that there is considerable historical doubt whether Jesus ever even existed is another issue.) Here is the ultimate root of anti-semitism.

As western society became more advanced there grew a need for bankers to facilitate economic growth. However, the bible says it is a sin to loan money with the expectation of collecting interest. How do you develop a banking system then? Well, the jews were already going to hell so let them be the bankers of society! Thus jews became prosperous not only from their cultural predilection to be successful, but also because of circumstances thrust upon them by the societies they lived in.

This prosperity matched with the strong jewish cultural identity which they held on to through 2000 years of diaspora, fueled both economic and religious resentment and of course conspiracy theories, which led to pogroms and atrocities throughout Europe over the ages, culminating in the holocast a mere lifetime ago.

The holocast led directly to the creation of Israel, which arab muslims saw as an injustice to the palestinians. Thus the modern anti-semitism of the arab world. In fact muslims were far friendlier to jews historically than christians were.

But essentially, the hatred is irrationally grounded in religion and economic resentment.