Learn Finnish: He ostivat uuden trampoliinin lapsilleen. - They bought a new trampoline for their children.

Because the children belong to the subject of the sentence (He). When a third-person subject owns the noun in the sentence, you must drop the pronoun (hänen or heidän) and just use the possessive suffix. If you added heidän here, it would mean they bought the trampoline for someone else's children!

I learned the third-person possessive suffix is -nsa or -nsä. Why does lapsilleen end in -en?

While lapsillensa is technically correct, Finnish has a secondary third-person suffix rule: if the case ending finishes with a short vowel (like the -e in the allative -lle), you can just lengthen that vowel and add -n. This makes lapsilleen, which sounds much smoother and is far more common in practice.

Why do uuden trampoliinin end in -n instead of being in the partitive case?

Because a trampoline is a single, countable item and the action of buying is complete. This requires the total object case (often called the genitive-accusative). If you used the partitive (uutta trampoliinia), it would sound like the act of buying was somehow interrupted or ongoing.

What case is being used for the word lapsilleen?