My understanding is that animal sacrifice was centered around the Temple in Jerusalem. The Romans destroyed this Temple at the end of the First Jewish Revolt in 70 C.E. Without access to the central holy site where sacrifices were to be made, Judaism ceased to become a sacrificial religion. This eventually led to modern-day rabbinical Judaism, which is based upon local synagogues rather than one central Temple.

Patpgh84 is correct that destruction of the Temple in 70CE ended animal sacrifice for all practical purposes, but the feeling that animal sacrifice was not necessary went back centuries. For example, in Hosea, which may be dated to the 8th century BCE, the prophet states “for I desire steadfast love and not sacrifice, the knowledge of God rather than burnt offerings.”

Ie Hosea felt that God did not want sacrifices, and simply offering sacrifices would not fulful the obligations of the people.

FYI it's currently the holiday of Shavuos, meaning observant Jews are not using electricity until Monday. As a result, some of the people that ordinarily answer these questions aren't currently available.

I'm a bit confused though... don't some Jews still sacrifice animals today? Or are Muslims the only Abrahamic faith that still sacrifices animals?

Actually, there are religious groups in Jerusalem right now who are prepared to continue sacrifice when the temple gets rebuilt. (This comment doesn't violate 20 year rule because these groups been around more than 20 years).

For a time, didn't they have human sacrifices/blood libel?