What he means is, we're wizards. We're capable of wielding magic which ... is sort of a broad statement, I know, but it comes down to casting spells and charms or curses. This extends to brewing potions, divination, et cetera. It's also a cultural marker for us, in that we have access to a world that's entirely hidden from non-magical people. A lot of wizards live entirely in this world, and it's your reality that seems entirely foreign to us. Well, some of them. My mother was a Muggle (non-magical) and I've always sort of had a foot in both.
It's usually something we keep very close to the chest, but apparently everyone's rather open here.