The answer is:
a) there will always be inadequate people who want someone to bully and/or make themselves feel better than. Anyone who is different in some way - Jewish, ginger hair, whatever, will get picked on by losers. In societies that were very uniform with no other visible ethnic minorities until recently, Jews stood out.
b) 2000 years of officially-sanctioned Christian antisemitism has had a huge impact on people, who have often been influenced by this without even realising. Until the 1960s, unbelievably, the official position of the Catholic Church was that all Jews (including current living ones) were to blame for the death of Christ. This leads to Jewish people now still being attacked as 'Christ killers'.
c) Medieval antisemitism forced Jews into certain jobs like moneylending as Jews were not allowed to own land or many other jobs. This increased resentment of Jews.
d) suspicion of the 'Other' and official Church antisemitism led to belief in various antisemitic tropes (hateful lies) such as the Blood Libel (that Jews kill Christian children to use their blood to make matzos for Passover) and Jewish conspiracy theories (the most notorious is the Protocols of the Elders of Zion, an early 20th century forgery which claimed that Jews were scheming to take over the world); this still comes out in all the 'Jews control the media / banks / whatever' crap.
e) The Israel/Palestine conflict has allowed a new facet to emerge, as anti-Zionism is used as a polite way of expressing antisemitism, by those who've realised that post-Holocaust, naked antisemitism sounds crude. Hence people who call Jews 'Zios' and blame all Jews globally for any actions of Israel they dislike eg the nutters who drove around at the weekend threatening to rape Jewish women and girls.
f) Because there are such a tiny number of Jews globally ( eg in the UK, only 0.4% of the population is Jewish), many/most people will not actually know any Jews (or even if they do, not realise this, as many Jews will try not to go public as Jews because of antisemitism). So in the absence of knowledge, people will believe any old crap about Jews.
That's a very potted summary. Suffice to say it has been baked into Western culture for 2000 years and also spread to other parts of the world more recently through propaganda so many people are unaware of it and don't spot it. It's unusual compared to other forms of racism as it's common among both those nominally on the far left as well as those on the far right although it manifests itself differently - the far left blame Jews for capitalism and see Jews as arch-capitalists (like Corbyn's infamous mural) as well as blaming all Jews for the Israeli-Palestinian conflict, while the far right see Jews as communists and to blame for Muslim immigration (yes really). It's all completely incoherent, but that's racism for you.