Learn Danish: Dette jakkesæt passer hende meget bedre.

Because the word for 'suit' is a neuter word (et jakkesæt). When pointing out a specific neuter noun, you must use the neuter demonstrative dette (this) instead of the common gender form denne.

Is jakkesæt a compound word, and does that affect why we use dette?

Yes, it is made of jakke (jacket) and sæt (set or outfit). In Danish, the gender of a compound word is always decided by its final part. Since it is et sæt (a neuter word), the whole word becomes et jakkesæt, which is exactly why we need dette here.

In English we just say 'fits her'. Do we ever need to use a preposition like til with passer in Danish?

Not when talking about clothing size or physical fit! Just like in English, you can put the pronoun right after the verb: passer hende (fits her). You would use passer til if you meant that the suit matches her style or goes well with her other clothes.

Why is 'much better' translated as meget bedre instead of mere bedre?

Bedre is already the comparative form of god or godt (good/well), meaning 'better'. Just like in English where you say 'much better' instead of 'more better', in Danish you use meget (much) to make the comparison stronger, rather than adding mere (more).