"This smart farmer is happy."
Translation:Hauʻoli kēia mahiʻai akamai.
"He" indicates that the thing following it is a singular (or general sometimes, I think) noun. So putting "he" in front of "hau‘oli" turns out into a noun that you are equating to the next thing. "He hau‘oli kēia mahi‘ai akamai" means "This smart farmer = a happy", if that means anything at all.
If the sentence begins with a base word and not a noun determiner (like "he", "‘o", "kēia", a number, etc.) then the base word is acting like a verb, in this case "to be happy". So we get the translation pairing that is above from this exercise.
If you left "he" off of "He wahine u‘i au," it would be treating "beautiful woman" as a verb and I don't know what that would mean either.
I guess the basic difference is that "hau‘oli" works better as verb (or adjective) and not so good as a noun, so it doesn't get "he" (which marks a noun). And "wahine" works better as a noun and not so good as a verb (or adjective) so it does get marked with a noun determiner (like "he" or others).
Kēia - this kēlā - that (away from the listener) kēnā - that (near the listener)
In second duolingo hawaiian lesson ("determiners") you can find such a description:
""Kēlā" and "Kēnā" both mean "that". The difference is kēlā refers to "that" which is away from the listener and kēnā refers to "that" which is near the listener. Cultural note of interest: Hawaiians are keenly aware of space and time.
So in terms of distance from the speaker, remember this order: kēia - kēnā - kēlā. (this - that (near the listener) - that over there)"