I can only provide some kind of a “qualified guess” I am afraid. For me, the only difference between these two sentences is the point in time when the answer is given.
“I got it.” might imply that someone has received anything some time ago. (past).
“I get it.” could perhaps convey the meaning that the answer is more related to the very moment of speaking.
However, I myself am not sure if this attempt to explain it is halfway right. As far as I have seen this expression so far it was always “I got it.” That is to say one can answer this way just a second after having received anything. But again "I get it" doesn't sound really "native" to me....
May I asked you what the context is where you found “I get it”? Maybe this could give a hint what might be meant.