As per my readings on Buddhism it doesn't accept the notion of soul or Atman, but at the same time it accepts the law of Karma. How does Buddhism explains rebirth and law of Karma?
All of our experiences in life ( The Five aggregates) are subject to the Three Universal Characteristics, one of these Characteristics is Impermanence. Every mental or physical phenomena ( Four Foundations of mindfulness) we experience through our six senses is constantly fleeting, in a state of change, this is " rebirth" or more precisely,this is " Birth". It is one experience changing into the next. Without impermanence, there would be no rebirth or birth of a new experience, in fact there would be no existence of anything!!. So...we all can validate this teaching of the Buddha.
As for Karma, i feel the teaching on Dependent Origination and The Noble Eightfold Path helps to see and understand " cause and effect" Our actions expressed through our thinking, speech and bodily actions condition the next moment, and in doing so, creates the possible conditions for our Karma to manifest from our previous or past actions ( whether that be this life or many others.
Once you study and understand the above teachings i have mentioned, you will see, experientially, the correlation between a wholesome action and its corresponding wholesome result, also you will see the corresponding result of an unwholesome action. This is universal law.
As the Buddha said, you don't have to believe anyone's word, you can see all of the Buddhas teachings for yourself!!
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:
We can considerate the fact that Siddharta Gautama was a person born in an Hindu environment, in which the rebirth principle is heavily present. Therefore his teaching is in the continuation of the education he could have during his childhood, just like Jesus was born as a Jewish.
Some believe that he truly believed in reincarnation, but in my opinion, we can take this aspect off the core of his thoughts that revolve around the personal experience.