In short: "Pay yourself first" is the principle that saving is the first expense on payday, not whatever is left at the end of the month.
Simply put: Most people save "what is left", and usually nothing is left. The idea is old — from George Clason's The Richest Man in Babylon (1926). It works best automatically: a transfer to a savings account on the day your pay arrives. Madrian and Shea (2001) showed that when saving is the default, many more people save — we rarely undo what is already set up. The habit matters more than the amount.
Example: Pay of €1,400. On payday 10% — €140 — goes automatically to a savings account. Over a year that is €1,680, without deciding every month.
Watch out: Myth: "I will save when I earn more." As income grows, spending usually grows too; the habit is built on small amounts.
Source: Madrian & Shea, "The Power of Suggestion", Quarterly Journal of Economics, 2001 — https://www.nber.org/papers/w7682