Preposition phrase after slogan

"Away from my hometown" is a definite clause that omits the subject and predicate and is used to modify five years.

This sentence (exactly, it is a prepositional phrase) is:

Five years after leaving my hometown.

This is a common form of ellipsis.

To add:

This ellipsis is usually used in the subject of the following main clause when it is consistent with the subject of the prepositional phrase as an adverbial. In other words, generally speaking, the following sentence should be

For example, five years after I left my hometown, I was seriously homesick.

The subject is also me.