They can offset donations against their tax bill.
So, if I give Archewell $50,000, I pay tax on $50,000 less of income.
Instead of the government getting (say) $5,000 in taxes from me, Archewell gets $50,000. The trouble with Delaware-incorporated foundations is that only $2,500 needs to be spent on actual charitable concerns. $47,500 can be spent on (say) clothes and security bills for the foundation's founders.
Why would a donor do this, you ask - why would a donor not just pay $5,000 in taxes? Better than $50,000 in donations, surely? Two reasons: (1) sometimes, there is a genuine desire to give to a given cause. It's an established cultural "duty" to give charitably in return for low taxes. Culturally Americans hate taxes and prefer to be in charge of deciding where their money goes. Socialism is a dirty word, and the government can withdraw from certain public-good causes because they've offered a tax incentive to ensure these causes will be privately funded (2) more frequently, especially in the circles Harry & Meghan are now in, because the donor gets something back. Publicity, association, a reputational lift - you scratch my back, I'll scratch yours. It's business under the cover of "charitable giving".
This is where Archewell will get into trouble very quickly. It, and the two people associated with it, have nothing to offer to donors. Nothing. H&M would have liked to position it as the American arm of the royal family, for American donors to benefit from (royalty-by-association). H&M have ensured, by their actions and their words, that that's not happening.
Archewell will very quickly be starved of funds. It's dead in the water.
ETA: a good comparison would be the Obamas' foundation. People will ALWAYS want to be associated with the Obamas' good works. They genuinely good; he was a much-loved President; they're trustworthy as far as these things go; they're reliable; their chosen causes and concerns are relevant, not specific to their personal gripes, they're coherent; they still exert influence. None of these things apply to Meghan and Harry.
There's a video clip of Ivanka Trump hobnobbing at an IMF or UN meeting. She looks the part, she's THERE, she smiles. She approaches a group of 3 women who are talking in a small huddle, one of whom is Ursula van der Leyen. She says something. Ursula looks at her blankly for a second and goes back to the conversation she was having (it didn't look particularly important, she had a glass of wine in her hand). Ivanka Trump looked chastened, awkward, embarrassed. That's what Meghan is in the world of philanthropy: she's got the clothes, she's got the plastic surgery, she's in the room. But that's it. Harry has a bit more, but he's the guy with a reputation that precedes him - you quickly find someone else to talk to when it looks like he's coming your way. Ivanka Trump has clearly realised her limitations, she's gone back to her natural home of just being rich. Meghan & Harry have no such self-awareness.