The usual budget works like this: you get paid, you spend, and you save what remains. Because what remains is a residual, there is almost always little of it. Paying yourself first inverts the order: the transfer to savings leaves on payday, and you live on the rest.
It works for two reasons and neither is arithmetic. The first is that it removes the repeated decision, which is where present bias does its work. The second is that spending adjusts to available money almost by itself, a fairly robust phenomenon: most people who automate 10% do not miss it three months later.
The advanced version is to escalate it: raise the automatic percentage every time income rises, before habit claims the increase.