This idiom or expression means:

A support.
Supporting material.
Available influence or power.

When someone says “He doesn’t have a leg to stand on,” they mean something like “He doesn’t have enough data to support his theory,” or “He doesn’t have an alibi to help him in his defense,” etc.

This idiom or expression has to do with:

Offering an excuse or explanation.

When you say: What do you have to say for yourself? you mean:

What’s your explanation?
What do you have to say in your own defense?