Alina is right. The verb is "von etwas träumen", i.e. the verb requires the preposition "von". Now, if you'd always dreamt of cats, you could say "Ich habe immer von Katzen geträumt". However, if you haven't got a simple noun (like "cats") as an object, but a verb-noun construction (here: to become someone/something - jemand/etwas werden), "davon" (literally: thereof) works as a kind of pronoun, which represents the object and which can be inserted where it belongs: between the conjugated part of the verb (habe) and the past participle/main verb (geträumt). Here is an example to show you what "davon" does: Als Kind wollte ich ein Star werden. (I wanted to become a star when I was a child.) Ich habe davon geträumt. (I dreamt of it/thereof.) And yes, this works with other prepositions as well: "damit", "darauf", "darunter", "daran", etc.". The great thing is that these "pronouns" don't change with case and/or gender. ;)

Yes, it is needed. It means "of" (träumen "von" = dream "of"). In such construction of sentences you use "davon" (instead of "von"). I hope German native speakers can/will explain it better.