How does the religion of judaism explain the existence of man?

You would get a very similar answer if you asked about the origin and purpose of man on Earth based on Judaism, Christianity, or Islam. All of these religions would agree that man was created by God and placed on Earth in order to worship God. Exactly why God wishes to be worshiped is not explained, but we are left to imagine either that God is very egotistical, or that He has some purpose which is beyond the comprehension of mere humans. Take your pick.

Judaism also teaches that we are here to make the world a better place, called Tikkun Olam.

Answer:Judaism teaches that God created us because of His desire for kindliness. He wishes to reward us, based upon our proper use of free-will in the tests of life. The place of that reward is (mostly) in the afterlife, since the very righteous deserve more than the limitations of this world would make possible.