Well, first of all you have the cities of Pompeii and Herculaneum, both of which are located very close to Vesuvius and both of which were (and are) buried under metres of volcanic ash and rock.

Even if there were no other evidence than that, we can put a fairly accurate date on when the eruption happened based on the artefacts that were found in the rubble. Coins are very useful in this regard. Coins with (e.g.) Vespasian’s portrait on them point to a date no earlier than the reign of Vespasian, for example.

In addition to this very obvious physical evidence, we also have a huge amount of literary evidence, not least the writings of an actual eye-witness to the eruption: Pliny the Younger. We have two letters that Pliny wrote to the historian Tacitus describing the eruption in 79 AD – 6.16 and 6.20.

Here’s the kind of thing he wrote: “Meanwhile on Mount Vesuvius broad sheets of fire and leaping flames blazed at several points, their bright glare emphasized by the darkness of night.” Full translation here: http://www.u.arizona.edu/~afutrell/404b/web%20rdgs/pliny%20on%20vesuvius.htm

That’s a basic answer. Let me know if you want me to go into more detail.