Learn Finnish: Ai jaa! Se oli ihana matka. - Oh, I see! It was a lovely trip.

Oli is the past tense ("was") of the verb olla (to be) for the pronoun se (it). This matches the lesson! If you wanted to say "It IS a lovely trip," you would use the present tense on: Se on ihana matka.

What exactly does ai jaa mean, and when do Finns use it?

Ai jaa translates roughly to "Oh, I see!" or "Oh, really?" It is a very common everyday reaction in spoken Finnish when you hear new information, used to show the other person that you are listening to their story.

I know hän means "he" or "she". Could I use it here instead of se?

No, in standard Finnish hän is strictly used for people. For inanimate objects, animals, or concepts like a trip (matka), you must use se (it).

Why are ihana and matka just in their basic dictionary forms?

Because this sentence uses the verb olla (to be) to describe one complete, singular thing (a trip). In Finnish, when you say "A is B" and B is a whole, countable thing, the words stay in their basic dictionary form (the nominative case).