This is an important question. While we today decisively reject the terrible things that people in the past thought and did, we should not shy away from recognizing what they did and why. And as Christians we must be honest about the role of religion and the church.
The "Middle Ages" covers a thousand years and the whole of Europe - and Jewish life varied considerably in different times and places - so this is going to be a bit general. If there are specific things you'd like to know more about, please do ask more questions.
Among the arguments used to justify anti-semitism at the time are the following:
- Supersession. The mantle of 'the chosen people' passed to Christians, and the old covenant has been superseded by the new.
- Christ-killing. The Jews were held collectively responsible for the death of Jesus: his blood was on their heads. This is related to supersession, because it shows the covenant being broken.
- General heresy. Anyone who did not share the Christian faith was not truly part of the community. This was used to justify various restrictions on holding public office, receiving university education, entering into mixed marriage, owning property, testifying in court, entering into contracts, etc. In addition it was common to think that Jewish people were particularly devious and untrustworthy, more so than regular Christian heretics.
- Usury. New Testament prohibitions on usury were invoked as a sign of Jews being evil (exploitative of "honest Christian folk").
- Evil magic. Again related to Christ-killing and general mistrust, some thought that Jewish people were in league with the devil, who granted them magical powers which they used to the ruin of the local community - especially by spreading disease. The infamous blood libel (that they murdered and ate Christian babies) was seen as a Satanic parody of the Eucharist.
As you can see, the basic theological root is the idea that by rejecting Jesus, the old covenant was abrogated and the Jews were no longer the chosen people. However, medieval anti-semitism is not just about theology: it also has a lot to do with not liking people who are different. There is also a tremendously important political and economic angle. It would be too simplistic to say that the religious or political establishment was solely responsible, or that it was all about popular prejudice. These aspects all strengthened and supported each other.
Meanwhile, you also wanted to know about the Old Testament being written in Hebrew. This is true. But there was a long period when knowledge of the language was pretty scanty - certainly among everyday clergy. The standard circulating text was in Latin, which was also the language of liturgy, scholarship, law, and public administration. The Greek "Septuagint" was also important - again, though, a bit more specialised. Most people would not have encountered any Hebrew text. But even for those who did, the idea of succession of covenants allowed them to receive the Old Testament scripture as part of Christian patrimony. They could praise the great virtue of Abraham, Moses, or David, even speaking in glowing terms of the magnificence of the Temple and the detail of Jewish religious life, while still rejecting Jewish people who were born in years AD instead of BC.