Learn French: Il a dit qu'il partirait le lendemain. - He said he would leave the next day.

Because this is past reported speech. When the introductory verb is in a past tense (a dit), an originally future action must shift to the present conditional (partirait) to express the "future in the past." If this were direct speech, the original quote would have used the standard future tense: "Je partirai demain" (I will leave tomorrow).

Why do we use le lendemain instead of demain for "the next day"?

In past reported speech, time expressions must shift to reflect that the original "tomorrow" is now in the past. The word demain (tomorrow) always turns into le lendemain (the following day) when recounting what someone said previously.

Could I use the verb quitterait here instead of partirait?

No, because the verb quitter strictly requires a direct object, meaning you have to state exactly what or whom is being left (for example, il quitterait la maison). Since no specific place is mentioned in this sentence, you must use partir, which works perfectly on its own without an object.

Because there is a que (qu'il), shouldn't the following verb be in the subjunctive?