Learn Spanish (Latin America): Explicaron que el drenaje urbano mejoraría. - They explained that the urban drainage would improve.

This is an example of reported speech. The original speakers likely used the future tense, saying El drenaje mejorará (The drainage will improve). Because the reporting verb explicaron is in the past, the future tense shifts to the conditional tense (mejoraría) to express a future action from a past perspective.

Since explicaron que introduces a second clause, why don't we use the subjunctive?

Verbs of communication like explicar (to explain) and decir (to say) just report information. They only trigger the subjunctive when they are used to give commands or influence behavior. Here, they are just stating a fact about the drainage, so we stick to the indicative mood.

Does the verb mejorar need a reflexive pronoun here, like se mejoraría?

No, mejorar can be used on its own to mean that a situation or thing gets better, just like the intransitive use of improve in English. The reflexive form mejorarse is generally reserved for people recovering from an illness.