Learn Hardcore Norwegian: Han kjenner seg selv bedre nå.

kjenne – to know in the sense of be familiar with, be acquainted with, know personally/through experience.
- Han kjenner seg selv bedre nå. = He knows himself better now (he is more familiar with himself, with who he is).
- Jeg kjenner ham. = I know him (personally).
vite – to know facts, information.
- Han vet svaret. = He knows the answer.
- Jeg vet at han kommer. = I know that he is coming.

In Han kjenner seg selv bedre nå, we’re talking about deeper familiarity/understanding of oneself, so kjenner is correct; vet would sound wrong here.

Why do we say seg selv and not just seg or ham?

Seg is the reflexive pronoun for han / hun / de (“he / she / they”) and it refers back to the subject of the sentence.

Han kjenner seg … = He knows himself … (referring back to han)
If you said Han kjenner ham, it would mean He knows him (some other male person, not himself).

Selv is added for emphasis or clarity, and together seg selv is a very common way to say “himself / herself / themselves” in the reflexive sense:

Han kjenner seg selv. = He knows himself.
Hun kjenner seg selv. = She knows herself.
De kjenner seg selv. = They know themselves.

So seg selv is the natural way to express “himself” when the object refers back to the subject.