Learn Finnish: Mä jatkais matkaa väsymyksestä huolimatta. - I would continue the journey despite fatigue.

The word huolimatta (despite/regardless of) is a postposition that specifically requires the noun preceding it to be in the elative case (-sta/-stä). Therefore, väsymys (fatigue) must take the -stä ending when used with it.

How does the base word väsymys change into the stem väsymykse-?

Nouns ending in -us/-ys that are derived from verbs or adjectives (like väsyä -> väsymys) belong to a noun class where the stem changes to -ukse- or -ykse- before case endings are added. Thus, väsymys becomes väsymykse- + stä.

Why doesn't the verb jatkais have a personal ending like -in for "I"?

This is a very common feature of spoken Finnish. In colloquial speech, the personal endings of the conditional mood (-isi-) are frequently dropped. The standard Finnish would be minä jatkaisin, but here the spoken pronoun mä is enough to tell us it's the first person.

Why is the object matka (journey) in the partitive case (matkaa)?

The verb jatkaa (to continue) generally requires a partitive object. This is because continuing something is inherently an ongoing, continuous process rather than an action that has been brought to a completed endpoint.