Learn Dutch: De kat blijft in de boom klimmen. - The cat keeps climbing in the tree.

Yes, blijven literally translates to "to stay" or "to remain". However, when you pair it with an infinitive verb at the end of the sentence (like klimmen), it means to keep doing that action. So blijft ... klimmen means "keeps climbing".

Why is klimmen at the very end of the sentence?

In Dutch, when you have a conjugated verb (blijft) and an infinitive verb (klimmen), the conjugated verb takes the second position in the sentence, and the infinitive gets pushed all the way to the end. That is why the location in de boom is sandwiched between them.

Could I say De kat is in de boom aan het klimmen instead?

You could, but the meaning is slightly different. Using is ... aan het klimmen simply means the cat "is climbing" right now. Using blijft ... klimmen emphasizes that the cat keeps climbing over a sustained period of time.

Can I use the verb houden for "keeps" in sentences like this?

No, you cannot. While houden means "to hold" or "to keep" (like keeping a physical object), it is never used to say you "keep doing" an action. For continuing an action, always use blijven plus an infinitive verb.