Learn Italian: Luca ha detto che al liceo eri molto studioso. - Luca said that in high school you were very studious.

This is an example of indirect speech. Because the reporting verb ha detto (said) is in the past, the tense of the original quote shifts back in time. Luca's original thought was likely in the present tense: "You are very studious" (sei molto studioso). When reported in the past, the present tense sei backshifts to the imperfetto tense eri.

In English, we can just say "Luca said you were very studious" and drop the word "that." Can I drop che in Italian?

No, you can never drop che in this type of sentence. While English often leaves out the conjunction "that" in indirect speech, Italian strictly requires che to connect the main clause to the reported statement.

Why does the sentence use al liceo to mean "in high school"?

Liceo is the Italian word for an academic high school. When talking about being at an institution like a school or university, Italian typically uses the preposition a (at/to). Combined with the article il, it becomes al, so you are literally saying "at the high school."

Does the ending of studioso change depending on who Luca is talking about?