Why is this word order correct, as opposed to putting the verb at the end of the sentence?
Frater meus habet unum filium.
This site supplied this quote.
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The word order is quite free in Latin. It is typical to put the verb last, but by no means compulsory. The example sentence you give has a correct word order, but it is not the only correct one. It would sound a bit more natural to put habet at the end.
The online learning site you link to seems to cut an awful lot of corners short. It can be a good tool to quickly get some idea of how Latin works, but if your goals are higher than that, I strongly suggest finding a better source and building your Latin on a more solid foundation. The site does not explain much and it does contain mistakes.
I have learned Latin from Finnish books, so, assuming you are not fluent in Finnish, I cannot suggest a book. I will therefore have to leave it to others to suggest good learning material.
Joonas Ilmavirta's answer is exactly correct.
I'll just add two things.
First, although word order is relatively free in Latin, the default position, as you rightly believe, is Frater meus unum filium habet. Changing a sentence's word order from the default adds a slight change in emphasis—here, emphasizing the fact that your brother has one son, rather than one ship, or one Gorgon, or perhaps three sons. The emphasis would be even stronger if unum filium came first in the sentence.
The second point to consider is that adding unum does more than just adding the Latin equivalent of "a" (as in, "My brother has a son"). If that were all you wanted to do, you'd just say, Frater meus filium habet or, more likely, Frater filium habet. Adding unum makes the sentence mean something more like, "My brother has one son" or "My brother has one son, and only one."
Of course word order depends a great deal on context. But in the absence of any context, the two most natural ways to write this sentence, it seems to me, would be
Frater filium habet ("My brother has a son")
Unum filium habet frater ("My brother has just one son").
The main point about word order, however, still holds: Both Frater meus habet unum filium and Frater meus unum filium habet are correct.