How does Buddhism explains rebirth and law of Karma?
The Kamma can be understood as a seed in put in your mind, what goes to the next life according to Buddhism is the mind, not a soul, the mindstream does not stop when the body dies.

There is a great video on Kamma by Ven. Yuttadhammo Bhikkhu on youtube, I recommend you watch it, it is full of details and at the same time easy to understand, I will paste the link:

http://m.youtube.com/watch?v=ih0moT0bvgM