A letter of intent isn't a binding contract. You contact your recruiter with a simple, straightforward email/call stating that at this time, an offer more suited to your life circumstances has been presented to you. They'll respond how they respond.
For what it's worth, while the previous generation seems to have this idea that you're beholden to whatever job you accept regardless of what offer you get afterward (I'm having flashbacks to my parents thinking my spouse should have stuck with an offer he'd accepted even when he then got a better offer before ever starting to work), you need to do what's best for you. You don't owe them anything. If they can't or won't match a better offer you get, you have no obligation to stick with it.
As the others wrote, a letter of intent is to indicate both parties are serious and serves as a starting point for negotiation. But it is not binding for either party.
If I may ... don't sign anything you don't understand. If you don't fully understand what you are asked to sign, research it or get advice first. If you are still unsure, get the advice of a lawyer.
You are now a highly educated professional. If things go sideways, claiming you didn't know what you signed is not going to fly.