As a Fighter with an above-average Intelligence score, I've chosen to take the Magic Initiate feat (PHB, p. 168), and as my 1st-level wizard spell, I've chosen to learn burning hands, which I'm now able to cast once per day as if I were a Wizard. This fulfills the requirements specified by the Boon of Spell Mastery (DMG, p. 232), which reads:
Choose one 1st-level sorcerer, warlock, or wizard spell that you can cast. You can now cast that spell at its lowest level without expending a spell slot.
But there's a snag, in the fact that this fighter does not have spell slots to begin with—they are not an Eldritch Knight, and no part of the Magic Initiate feat specifies that they gain any spell slots. They weren't casting burning hands using a spell slot in the first place, so saying they "can now cast burning hands without a spell slot" doesn't mean anything.
Does this Boon benefit this character at all or is it useless to them?
The character would benefit from the boon
Magic Initiate says:
As you correctly say, Magic Initiate does not grant you spell slots, only (from this bullit) the ability to cast a single spell. So, you are already casting this spell without using up a spell slot. However, Magic Initiate only allows you to cast the spell once per long rest. Upon receiving the boon, you would be able to cast it as many times as you wanted. So there would be a benefit there.