1. Knowing Trump, he'd rather "appropriate" real estate rather than pay for it. Even though he'll argue there's a very large federal deficit, he'd find the money to first try to buy Greenland. But that won't happen.
2. So, using a falsehood of a crisis, maybe he'd send the military into Greenland to "protect" it -- from China, who knows? And using American imperialism (if not Israeli MO), the troops would stay in Greenland indefinitely.
3. I don't know how he'd do it, but whatever wealth Greenland has, Trump would want a taste of the action. Leasing out land for excavating the soil for minerals, perhaps even clean, fresh water, would be another source of income for him.
4. And on a non-pecuniary note, I honestly believe Trump likes (a lot) the concept of taking over another country, making its residents part of his kingdom. Greenland would now take orders from Trump, not themselves, not Denmark.
5. Trump likes to prove that he can do something that everyone tells him would be impossible to do.
Exhibit A: The Dobbs decision. Everyone told Trump that it would impossible to overturn Roe and he did it.
Exhibit B: More than 30 years ago, Trump attempted to use eminent domain to get ownership of a small property near his casino in Atlantic City. (He wanted the lot to develop a parking garage for limos to his business.) He first offered the owner, a widow, a pittance of the value of the property, which she turned down. Trump took her to court again and again. He was told it wasn't worth the time and money, but he believed he could wear out the widow. Not exactly. Trump had to abandon his plans and was chagrined that he lost to an elderly woman. https://en.wikipedia.org/wiki/Vera_Coking_house