Jews don't believe Jesus is the Messiah because he didn't fulfill all of the prophecies, or some say, any of them. However Christians believe he will fulfill them when he comes back.
More deeply, however, is the problem that Christians see Jesus as G-d, whereas Jews see G-d as formless and without a body. In Numbers it says that G-d isn't a man, and the references to G-d's hands and other body parts is taken as metaphorical.
I don't know whether Jews look down on bastards (people without a father), but they do look at Jesus as this, in the technical non-derogatory sense of the word. That is, he had a father of some kind, but the father didn't stay with the family, or wasn't Joseph and didn't marry Mary, therefore he is a bastard. "the illegitimate offspring of unmarried parents" But that rabbi may have meant it in a derogatory way; who knows, I'd have to see the clip.
Other Jews are skeptical that Jesus even lived. They wonder if the Gospel stories are just made up because they seem fantastic. I'm not sure why they would find them any more fantastic than the stories in The Tanach, though.
I think Jesus was a good man, but not G-d. He taught well, if we go by the gist of what he said. He was called rabbi by some of his followers. It's hard to argue with most of what he said because it was good, but a lot of what became the religion of Christianity is anathema to Judaism. Much of this originates with Paul and his epistles.
Edit to that last part that I struck through: I don't know that Jesus taught from God or was very righteous. He seemed to be a mentally ill person, with a fascinating philosophy and way of teaching, but not necessarily good, and not very Jewish -- certainly he was distinguished from the Pharisees, and modern Jews descend from those.