Learn Finnish: Hän tapaa vaellella metsässä, kun hän etsii inspiraatiota. - He tends to wander in the forest when he seeks inspiration.

This is the habitual use of tavata. When you pair it with a first infinitive (here, vaellella), it acts as an auxiliary verb meaning "to usually do" or "to tend to do" a customary action.

Why use vaellella instead of the basic verb vaeltaa (to hike/wander)?

Vaellella is a frequentative verb. The -ella suffix modifies vaeltaa to mean doing the action aimlessly, casually, or repeatedly. He isn't hiking a specific trail from point A to B; he is just wandering around.

Why is inspiraatiota in the partitive case?

The verb etsiä (to look for / seek) is a partitive verb. The object you are looking for must be in the partitive case because the action of searching is an ongoing process without a guaranteed, finalized result.

Since the subject is the same in both clauses, could we drop the second hän and just say kun etsii?

No, standard Finnish requires you to keep third-person pronouns (hän and he). You can generally only drop the subject pronoun for the first and second persons (minä, sinä, me, te).