When saying "there it is," a person usually means that something being sought for has at last been found, as the question notes. Most often, that something is a concrete, tangible object:
Alice: I can't find my cellphone. Have you seen it?
Bob: Have you checked in the refrigerator?
Alice: (opening fridge door) Why would it be in .... Ah, there it is.
Dave: Fred's directions said we need to turn left on Maple Avenue to get to the party. Do you see any Maple Avenue?
Charlene: (pointing to a distant street sign) Yes, there it is.
Dave: Wow, good eyesight.
However, the sought after subject can also be a more abstract or figurative thing, such as comprehension. The question also mentions this.
Louis: I don't get it. What happened to the baker?
Carrol: Well, you see, the Snark was a Boojum after all.
Louis: Oh. There it is.
In this case, the response is often given in a drawn-out, almost sing-song voice: Oooooooh, therrrrre iit iiiiis.
One additional meaning expresses that a speaker or writer has just laid out a detailed explanation, and is providing a summary.
Professor: So there it is, class. The cycle of cellular division involves interphase, prophase, prometaphase, metaphase, anaphase, and telophase.
Students: (half applaud, the other half are startled awake.)
Finally, a meaning given by Collins and Oxford says that "this is the state of affairs, this is the situation." In other words, whether one likes it or not, and there is nothing one can do to change it.
But there it is: you can't wait now till spring; and you can't go till the reports come back. (J.R.R. Tolkien THE LORD OF THE RINGS)