The word earn is implied but omitted from the end of the sentence. Adding it might help make the structure clearer:
He earns more in a month than I could hope to [earn] in a year.
- The speaker uses "could" because he or she is describing a hypothetical situation. Grammatically, instead of "can hope" (indicative tense), the speaker invokes a hypothetical with "could hope" (conditional tense).
- The second instance of "earn" is understood by the reader because it was already stated at the beginning of the sentence. Similarly, the word "to" in the following sentence is assumed to introduce an implied verb "to go":
They invited me to go, but I didn't want to.
Your original sentence is fine as is, but here are some alternative renderings of the same idea:
He earns more in a month than I could hope for in a year.
He earns more in a month than I could earn in a year.