Your coolness rank shows generally how well you've been doing in the online multiplayer during the current session. The session changes when the stages change, which is every four hours (despite that Judd's tip claims that it resets daily).
There are four different ranks. The highest number of flags you can have is nine, and the lowest is zero. Winning while your rank is low will gain you more flags than winning while your rank is high, and losing while your rank is low will cost you less flags than losing when your rank is high. Your coolness rank can go above 9.0 (I've gotten it as high as 44.0), but no more than 9 flags will be displayed.
The only thing you can receive from your coolness rank is coins, which will be given by Judd the cat in the plaza. You can receive coins once per coolness rank per session, for a total of 1,400 coins. When you talk to Judd, he will include any unclaimed rewards for lower ranks that you haven't yet collected in that session. Rewards are based on your highest rank so far during that session. You will not receive any unclaimed rewards when the session resets (also despite what Judd's tip implies).
Here are the ranks and corresponding rewards:
- Chill - 0 flags - no reward
- Toasty - 0.5 to 3 flags - 100 coins
- Smokin' - 3.5 to 6 flags - 300 coins
- So Hawt! - 6.5 to 9 flags - 1,000 coins
The text associated with each of these ranks depends on localization; these ones are for the American localization ("On Fire!", which you mentioned, is the European equivalent to "So Hawt!" in the European localization).