Learn Hardcore Afrikaans: Sy gee nuttige advies en hou by haar belofte om te help. - She gives useful advice and keeps her promise to help.

In Afrikaans, attributive adjectives (the ones that come before a noun) usually take an –e ending. So nuttig becomes nuttige when it modifies advies. If you used it predicatively (after a linking verb), you’d say:
Die advies is nuttig.

Why isn’t advies plural like English “advices” or “pieces of advice”?

Just like English advice, Afrikaans advies is a mass (uncountable) noun. It doesn’t normally have a plural form. If you want to refer to individual items, you’d say stukke advies (lit. “pieces of advice”).

What does hou by mean here, and how is it different from hou van?

hou by is a phrasal verb meaning to stick to or to adhere to something (e.g. hou by jou besluit = “stick to your decision”).
hou van means to like (e.g. Ek hou van aardbei-yoghurt = “I like strawberry yoghurt”).

Why can’t I just say hou haar belofte without the by?

Without by, hou haar belofte would literally be “hold her promise” (as if you’re physically grabbing it). The by is essential in hou by to convey the idea of “keeping to” or “honouring” a promise or agreement.

Why do we use om te help instead of just te help?

Afrikaans uses om te + infinitive to express purpose or intention (“in order to …”). So om te help means “to help” in the sense of “in order to help.” You cannot drop om in this kind of clause. (Note: after modal verbs like kan, you can say kan help without om.)

Why is there no article before nuttige advies or before haar belofte?

advies is uncountable, so it doesn’t take an article.
belofte is already made definite by the possessive haar, so you don’t add die or ’n.