Learn Russian: Что она думает обо мне?

The preposition meaning "about" is usually о. However, it changes to make pronunciation easier. It becomes об before vowels, and it becomes обо before certain words starting with difficult consonant clusters. The word мне starts with two consonants, so we say обо мне.

Why do we use мне instead of меня for "me"?

The preposition о / об / обо requires the following word to be in the Prepositional case. While меня is the form used for direct objects (the Accusative case), the Prepositional form of the pronoun я (I) is мне.

I've seen мне used to mean "to me" before. Is this the same thing?

You have a good memory! Мне is also the Dative case form, often translating to "to me" (like in Мне нравится - I like). For the pronoun я (I), the Dative and Prepositional cases happen to share the exact same spelling: мне. You can tell it is Prepositional here because it follows the preposition обо.

Why does the verb end in -ет (думает)?