It's important to remember that Judaism was a religion instituted by God, intended for God's purposes, but had been co-opted by man who began adding and removing laws as they saw fit.
So enter Jesus: He says he "fulfills" the law. What does he mean by this? Many people seem to think this meant that he added to, and finished creating all of the rules associated with the law, but I don't believe this to be the case. As Paul said in Romans 3:19-20:
Now we know that whatever the law says it speaks to those who are
under the law, so that every mouth may be stopped, and the whole world
may be held accountable to God. For by works of the law no human being
will be justified in his sight, since through the law comes knowledge
So if the law's purpose wasn't to primarily justify us, but to show us our sin, then why would Jesus simply come and add laws? (Indeed, he does do this at times, but I believe his point is to push people closer to the edge of despair when it comes to their own self-reliance, and push them to a point where they cry out to have mercy on them.) The answer is, he doesn't. He, being God, clarifies the law by showing the true expectation of the law, that our hearts would be turned towards God.
Now, as for fulfilling the law, he fulfills it by himself following the law. He finishes the expectations of the law by living under the law, and thus, he is the only one who manages, on his own merits, to escape the judgement of God. Now here's where it gets crazy, and why we believe in Christ in the first place: he then goes on to die the death of a sinner anyways, and in doing so, he becomes the sacrifice for the sins of those who have failed to obey the law, and thus he frees us from the law (though not himself).
Now, the temporal consequences of our sin will still remain with us. If I kill someone, then I am more than likely going to suffer temporal consequences for doing so, but the death of Christ is sufficient to pay the penalties for this on an eternal level, and my failure to match up to the requirements of the law are forgiven not solely on the basis that God forgives, but on the basis that Christ has already paid the penalty.
So Jesus reconciled himself to Judaism by following all that Judaism required of him.
As for his contemporaries trying to decide if he did or not, keep in mind that they had already co-opted the law and the prophets for their own gain. They added and removed laws to suit their fancy, and viewed Jesus as a glutton and a drunkard because he didn't follow their twisted version of the law. Nevertheless, the law was given by God, and the law, as given by God (not as twisted by the Pharisees and Sadducees) was fulfilled (obeyed) by Christ.