If you eat the same amount of calories everyday your body will adjust by lowering your metabolism. That is why dieting often leads to gaining more weight then what you lost. The cheat day prevents this from happening.
Are you sure metabolism works like that? A quick Google seems to say that is a myth.
I think dieting can often lead to weight gain due more to user error than body-trickery.
Yea it's because people are terrible at judging calorie content and physical activity for >99% of it (severe metabolic problems excluded). People that are shitty dieters will chronically under-count calories (and people who are "hard gainers" will chronically undercount). So you have weird ideas of calorie cycling etc as a way to "confuse" your body. Here's what matters for weight loss-
BMR / RMR - interchangeable terms for this point. Basically the calories to stand there and not die.
NEAT - non-exercise activity calorie burning. Walking around, playing with a pencil while thinking -> fidgeting.
EAT - exercise based calorie burn.
TEF - The caloric burn of eating food. Something like 10% of food eaten is used to burn food eaten.
The NEAT part of it is surprisingly impactful. There can be an observable shift in the fidgety-ness of a dieter vs. naturally thin person vs naturally overweight person. Some people just move more. That gets dampened when you diet. It can double your BMR if you have a labored job.
Some of BMR/RMR is static, like brain thinking and heart pumping. Some of it is variable, based on your weight and body composition. Strictly speaking, big people take more calories to exist as big people just walking down a hallway whether it's muscle or fat. So if you diet hard, you take less calories to stay alive and you burn less calories moving. And you tend to be less fidgety. And your workouts are shit because people don't work out hard when they are dieting. Is this a "metabolism" shift? It's a calorie burn shift, so terms are different but the end result has the same tone to it.
The cheat day helps one of those things (the feeling like you are dieting, so your workouts aren't as crappy), and realistically hampers dieting more than it helps if you were just not doing a cheat day and pushed straight through willpower. As long as you are consistently and appreciably under your TDEE (add up everything above), you'll lose weight. Just know your TDEE also changes.