Learn Finnish: Varustauduttuamme menimme luolastoon. - After equipping ourselves, we went into the dungeon.

It uses the past temporal construction. We start with the verb varustautua (to equip oneself). We take its passive past participle varustauduttu, add the partitive ending -a, and finally attach the first-person plural possessive suffix -mme (because we did it). This gives us varustauduttuamme (after we equipped ourselves).

If we equipped ourselves, why does varustauduttuamme use the passive past participle?

That is the strict rule for the past temporal construction: it always uses the passive past participle stem, even for active sentences. The person who performs the action isn't shown by an active verb form, but rather by the possessive suffix at the very end of the word (here, -mme for we).

What is the difference between luola and luolasto?

The suffix -sto or -stö turns a noun into a collective noun, meaning a group or network of that thing. Just as kirja (book) becomes kirjasto (library), luola (cave) becomes luolasto (a network of caves, or a dungeon). The ending -on is the illative case, meaning into.

Could we change the word order and put varustauduttuamme at the end of the sentence?

Yes, Menimme luolastoon varustauduttuamme is perfectly correct. Placing the temporal construction at the beginning of the sentence simply emphasizes the timeline of events: first we got equipped, then we went into the dungeon.