How are those tied together? Glad you asked.
In the last paragraph of the Vesta post, I said we have samples of Vesta that fell as meteorites. As it happens, they had a sample of one of those rocks at the show! Here it is:
Other such meteorites have been found on Earth as well, and are generally referred to as HEDs, short for Howardite/Eucrite/Diogenite -- the three main types of these rocks.
So how do we know these meteorites were once part of Vesta? A couple of ways, actually, but they rely on spectroscopy. That's the process of taking the light from an object and breaking it up into individual wavelengths (using a prism or a finely etched grating, for example), essentially creating a rainbow with as many as hundreds of thousands of colors. The spectrum of an object can be affected by its composition, how fast it's spinning, whether it's moving toward us or away from us, and much more. Reading its spectrum is like reading its DNA.
In the 1970s, studies of HED meteorites (PDF) indicated their spectra were much like the spectra of Vesta, showing their compositions were very similar. Now, asteroids are pretty diverse, so finding similarities between an asteroid and a meteorite is a pretty good piece of evidence that the latter came from the former. As it happens, though, there are other asteroids with similar spectra to Vesta, and they are in similar orbits as well. This is what you'd expect if, say, Vesta got hit by some large object, dislodging big chunks which then orbited the Sun on their own. This is seen commonly in the asteroid belt, and groups like this are called asteroid families.
Some of the asteroids in the Vesta family are not apparently similar to each other, though. Spectra show that some are chemically similar to rocks that formed in something like a planetary crust, while others look more like rocks from a planetary mantle. HEDs show the same property, in fact. So how can that be?
The answer is that Vesta apparently was once hot enough to have had heavier minerals sink to its core and lighter ones float to the top. That's called differentiation, and it's happened on Earth as well. That's why we have an iron core and a rocky crust.
So when Vesta formed, it got hot enough that metals sank to its core and rocky minerals floated up to its mantle and crust. Over millions of years it got impacted by other asteroids, getting huge pieces knocked loose -- in fact, as I pointed out in yesterday's post, the south pole impact on Vesta was a real whopper, and could easily have created all the Vesta-like asteroids known. Remember, Vesta is the second largest asteroid (or maybe third, depending on the uncertain size of the asteroid Pallas), so it has material to spare.
As these asteroids spread out, some stayed in similar Vesta-like orbits. Others got blown out farther, and had their orbits affected by Jupiter's gravity. Some of those had their orbits altered enough by the mighty planet that their paths take them near the Earth. Over time they too suffered impacts, creating smaller objects that then have orbits that cause them to hit the Earth. We find those as HED meteorites. Perhaps some came directly from Vesta, and others from Vesta's violently-birthed daughter asteroids, but in the end we think they can trace their lineage back to Vesta.
So there you go. It's a bit of a twisted tale, but it all hangs together, as scientific stories must. We don't know all the details, of course -- there could be other asteroids similar to Vesta that produce some of these meteorites, for example -- but that's why we explore. We sent the Dawn spacecraft to Vesta specifically to help answer questions like that, to make a mineralogical map of the asteroid to find out exactly what it is and how it's put together.
And in the meantime, we have actual samples of asteroids simply sitting on the Earth's surface, waiting to get picked up and examined. Normally, planetary scientists have to wait years or decades for probes like Dawn to reach their targets, but sometimes nature is kind, and brings those targets -- or at least small pieces of them -- right here to us.