It means always save some money for yourself. Put money in your emergency savings, enough to hold you over for six months, put money in your retirement account, or your home down payment savings fund. Don't neglect to save. EVER. Living paycheck to paycheck is something no one can afford to do. It can be extremely difficult to save but even a few dollars will add up over time.
Are you trying to set up a budget to get your bills paid? B/c if you are, "pay yourself first" means you need to set aside money for YOURSELF, BEFORE setting aside specific monthly amounts for outgoing bills. If you set up a budget to get your debt paid but don't leave any room for walking around, pocket money? That budget is doomed to fail before it ever really gets started.
For example: Let's say you bring in 1500.00 a month and you have 20,000.00 in Credit Card debt you want to hurry up and pay off, so you decide to put as much toward it as you can. Your rent, utilities, car payment & insurance totals 1000.00 each month. Your gas & groceries cost 200.00. You THINK this leaves you 300 to put toward the credit card.
But....you forgot to PAY YOURSELF FIRST each month. You forgot the pocket money, the loan-a-friend-forty-bucks money, the go have dinner & drinks with friends money, whatever you may give to the man on the street corner carrying a sign, etc. All of these odds & ends fall in the budget category of "Paying Yourself First." So. Successful budget? Set aside a REASONABLE amount of carrying-around money along with all of your other budget categories.