I am told in a lecture that some particular meteorite is from Vesta. How can we know that a particular meteorite is from there?
Meteorites have distinct compositions and so meteorites with similar compositions can be grouped into families. There are, for example the iron meteorites, meteorites with lots of carbon and so on.
One family are called HED meteorites. They don't contain chondrules and show evidence of igneous processing (the rocks have been melted). The are differentiated. This suggests that they didn't form from the raw materials in the solar system, but formed on a larger body, and then were ejected into space. The look very like igneous rocks on Earth.
It is suspected that these come from Vesta. They don't come from the Moon or Mars (we know about the composition of these worlds and they don't match). Vesta is big enough to be differentiated, and in the right place to allow for pieces of rock ejected by impacts to be perturbed into Earth-crossing orbits. Observations by the Dawn probe are consistent with Vesta being the source of HED meteorites (By contrast, Ceres has a different composition and so is not a likely source), and there is a large impact crater (the Rheasilvia crater) that could have ejected a huge amount of rock into orbit around the sun. Some of the debris from this impact would still be falling to Earth, even after a billion years.
So Vesta fits the requirements. That isn't quite proof that these HED meteorites did come from Vesta, but it is a reasonable belief.