The majority of messages I got from women were usually 'Hiya.. lol' or 'Howz u, babe'. Some of them were more wordy, and decent, and referred to something in my profile - those were more interesting.
But generally, if I didn't fancy the woman who'd sent the message, I wouldn't reply. I used to, to begin with, with a polite 'thanks but no thanks' message, but received vitriolic responses so I started to just ignore them.
I think I ended up having two or three dates with women who'd messaged me first. My current girlfriend is one of those, and we're now talking about moving in together next year - so it can work.
As for the 'Meet Me' thing - well, in my case I swiped lots of them, because there were hundreds of women on there, some of them looked good. Some would like me back, and so I'd send messages to some of them. But just because there's a mutual match, it doesn't mean I'd message every single one of them, there could be dozens.
It's a numbers game. You can get very lucky, or it might never work. Any man who finds you attractive and interesting and yet loses interest because you messaged first - well, he's a dick.