When you're in a relationship, and you seem to care more about it, what does it really mean?

Too many times you here people say something along the lines of “I care more about you than you do me” or “I’m putting more into this relationship.” My question to those who have experience with this is what does that usually mean?

Doesn’t one person always have to care than the other? It’s not like it can be a tie. There’s no score when it comes to feelings.

Also, do you think it’s that person just being insecure?

All thoughts are welcome