Learn Russian: Мы долго бродили по лесу, несмотря на то, что шёл дождь. - We roamed the forest for a long time despite the fact that it was raining.

бродили is the multidirectional verb for wandering or roaming (from бродить). Since they are walking aimlessly around or throughout the forest rather than trudging in a single, specific direction, the multidirectional verb is needed. брели (from брести) would imply walking slowly or dragging one's feet along a single path.

Why do we use the preposition по with the dative case (лесу) here?

по plus the dative case is frequently used with verbs of motion to indicate movement spread across a surface or area (meaning around or through). по лесу means the wandering was distributed all over the forest, which pairs perfectly with the multidirectional verb бродили.

What is the literal construction of несмотря на то, что?

It translates literally as not looking at that, that... but is the standard Russian phrase for despite the fact that or even though. The preposition на requires the accusative pronoun то, which acts as a grammatical placeholder to introduce the actual clause starting with что.

Why do we use the verb шёл for the rain?

In Russian, precipitation goes rather than falls. We use the unidirectional verb of motion идти (past tense: шёл for a masculine noun like дождь) to describe rain or snow actively falling. It is imperfective here because it was an ongoing background action while they were wandering.