Before answering this question, I should disclose that I am not a scientist, a philosopher, so parts of the question I would delegate to scientists, such as evolutionary biologists. That said, there are basic approaches to the first part of the question “how to understand evolution”.
What I would do is carefully split the understanding into two parts: Natural selection and random mutation.
For natural selection, I would explain it as when things live longer, they are more likely to reproduce. An example you could give is a human who lives to ths age of 15 has a much less chance of reproducing as a human who lives to 30.
On to random mutation. I would explain random mutation as when things are born, or reproduced, they sometimes have things about them that are different than before.
So, you can then say that sometimes something will be born different in a way that makes it more likely to live longer. And since it is more likely to live longer, it will be more likely to reproduce. And since they are more likely to reproduce, they are more likely to pass on those things that made them more likely to live longer to those being reproduced/born.
That, in basic terms, is how I would explain it. Now on to proving it to be true. The word prove I’d probably avoid, because things known through science are not known through proofs (such as logical or mathematical proofs), but through inference to the best explanation (my tutorial on inference to the nest explanation goes over this in more detail). So, I would tell them that evolution is the best explanation for what we have seen in nature.
If they want to know why it’s the best explanation, that would start requiring more scientific knowledge on the subject, where I would probably recommend introductory biology books. Or, I specifically recommend Richard Dawkins’ “The greatest show on Earth” who goes through all the evidence targetted at the lay person: https://amzn.to/2JjkfaU
Thanks for the intresting question,