What happens to a space probe when it has finished transmitting?

Answer 1

Presumably you mean when it runs out of power to the point that it is no longer capable of transmitting at all. If it is, for example, Voyager, then it will continue to drift out into deep space for probably longer than the human species will exist. It might eventually fall into the gravity well of a black hole somewhere, but space is unfathomably vast, so it is probably just going to drift forever.

Answer 2

Once a space probe has finished transmitting data, it may continue to orbit in space if its mission allows. If it is no longer operational or out of fuel, it will likely become space debris, orbiting until it eventually re-enters Earth's atmosphere and burns up upon re-entry.