The thing is it's a bit complicated because it depends on the airline, not so much on EU regulations themselves. What you can do is bring only carry-on baggage within the limits that the airline allows - usually a backpack or a small bag, often within 40x20x25 cm or thereabouts. The trick is that almost all airlines allow a free "personal item", like a travel backpack or a bag, if you stuff it under the seat in front of you. The problem is that with checked baggage you pay extra, and most of the budget carriers like Ryanair, Wizz Air and so on don't give anything for free anymore from a few years back. If you want to avoid paying altogether, you need to manage to bring everything you need in your personal item backpack and that's it. It's not a secret solution, it's just that airlines have slowly started to charge money for stuff that used to be free.
If you're a frequent flyer or want an alternative, it's worth checking out memberships like frequent flyer programs or annual packages that some airlines offer - Ryanair has Plus, Wizz Air has the club, and so on. With those you pay once a year and they generally include free checked baggage (or often two) plus other benefits. It's not a free solution, but if you fly often at least you break even on it. Or, consider airlines that still offer baggage included in the ticket price - there are still some European ones that do it, even though they generally cost a bit more per ticket. In the end it depends on how many times you fly a year and how much you're willing to spend overall.