A wet sponge is really important in soldering. After you've been leaving your soldering iron sitting in its stand for a period of time, the iron tends to form oxides on its surface that will interfere with the heating process of transferring heat from the iron to the work piece.
And you can see that the tip doesn't look as shiny as it did right before soldering, and the reason is that oxide is forming. So what we need to do is remove that solder by wiping it on a sponge, a wet sponge. And by doing that, we remove that oxide layer and expose fresh, un-oxidized metal, and that'll be much more effective at transferring heat to the work piece.
This is a brand new sponge, so it's been compressed and it's a little bit thinner than it will be after we wet it. Place it a bowl of water, or under a faucet, the sponge expands and now it's a bit thicker than it was. And we don't really want it saturated like this; it's kind of, I guess, too swampy, if you will. We want to get it a little bit dry, so we give it a ring, but not too dry, just enough to leave a little bit of moisture when you squeeze it. Just like that, that's perfect. Just below that in our soldering iron stand. Now we can use this to clean the tip.