I've heard this phrase a lot too. It is most definitely possible to love other people without loving yourself. But I think the most important message to take away from that phrase is not that you need to fully love yourself before trying to love other people, but to value yourself as a human being in relationships.

Aka don't let people walk over you and don't let them take advantage of you. I know a lot of people who have gotten their self-worth shot to pieces from relationships where the person who is supposed to have their best interest in mind is the one constantly tearing them down.

The phrasing surely could use some improvement but there is quite the difference to being "whipped" and bending over backwards to try to please. Wanting to make your partner happy is one thing, but when it takes a toll on your own mental or physical health it is quite another.

As for loving yourself, that really all depends. It's having an innate ability to believe that you are a good person placed in sometimes bad circumstance and that even if there are parts of you that you don't like, you still value yourself as an individual. To find good points about yourself among what sometimes feels like a never ending ocean of negative feelings. Often, it's best to ask family and friends that you trust and love to help you find them for you. Sometimes you find it in odd talents or random abilities you've acquired over time. But most of it comes with combating bad thoughts and filling them with optimism. That a lot of what happens is situational and not a true reflection of yourself. It's hard and takes a long time to grow but most definitely worth it.