Learn Hardcore Icelandic: Vasaljósið virkar betur eftir að hún skipti um rafhlöðupakka. - The flashlight works better after she changed the battery pack.

Vasaljósið = vasaljós (flashlight, neuter noun) + the suffixed definite article -ið (“the”).
So it literally means the flashlight. Icelandic usually puts “the” at the end of the noun like this:

hús → húsið (the house)
vasaljós → vasaljósið (the flashlight)

What grammatical form is virkar?

Virkar is the present tense, 3rd person singular of að virka (“to work / function”).
It agrees with the subject vasaljósið:

ég virka (rare/odd with this verb)
það virkar (it works)

In this sentence it’s “works/functions (now)”.

Why is betur used, and what exactly is it?

Betur is the comparative adverb of vel (“well”).

vel = well
betur = better
best = best

Because it describes how it works (an adverb), you use betur, not an adjective form.

Does betur require any special word order or extra words (like “much”)?