Learn Russian: Здесь нет автобуса номер шестьдесят.

It is a compound word made of шесть (six) and десят (an old form of the word for "ten"). It literally translates to "six tens". Notice that it keeps the soft sign (ь) in the middle after шесть, but unlike the word десять, it does not have a soft sign at the very end.

Since нет means "there is no", shouldn't the whole phrase be in the Genitive case? Why is it just автобуса?

Only the core noun that is actually missing or absent takes the Genitive case. Here, the missing thing is the bus, so автобус changes to автобуса. The phrase номер шестьдесят (number sixty) acts as a fixed label or name for that bus, so it stays in its default dictionary form (Nominative case).

Could I just say Здесь нет шестидесяти автобусов instead?

That would completely change the meaning! If you use the number to show quantity (and put everything in the Genitive), it means "There are not sixty buses here." Because we want to identify a specific route, we use the noun plus the word номер (number) followed by the flat number: автобуса номер шестьдесят.

I learned that нет just means "no". How does it mean "there is no" here?