Like someone else said, to 'water something down' is to dilute it. Sometimes it has a negative meaning, like if you are at a bar and say 'the drinks here are watered down', then you mean that there's not a lot of alcohol / flavor in the drink. In other contexts it can have a more neutral meaning, like if you're making soup or sauce, you might want to water it down to make more, or if the flavor is too strong, or it's too thick.
In the context of the song, she says 'you know me better than that, you know i hate it like that, it really waters me down' so she's saying, when this person treats her this way, it dilutes her feelings, it makes her emotion less strong, it decreases her energy. The song starts by saying "never meant to be you, never meant to be me, never meant to be us", so the whole song is sad and disappointed. She's letting the other person know that her feelings are weakening because of their behavior. It's about that feeling when you think things are going really well, but then the other person starts pulling away - the relationship gets 'watered down', it's not as strong anymore.
This isn't something that people say every day because it's more poetic, but if you were to say "our relationship feels watered down" or something like that, most people would know what you mean. Usually it's used to literally talk about putting more water in something. :) Also thanks for this question because now i'm listening to this song and it's lovely!