The awards in BH act like the achievements in any other basic game. They are basically goals to achieve that give Bug Heroes that bit more replayability. Awards can be featured on your Game Centre, yet otherwise they are offline.
The Awards section in Bug Heroes in reached by pressing the Awards button below "Coliseum." Once there, there are three categories: Overall, Heroes and Scrolls. If you have Game Cantre, then an extra button will appear beside Scrolls with the Game Centre logo.
In the Overall section, there are 15 awards that are achieved through different means that don't have anything to do with your heroes. So basically, these awards are based on your overall play. For example, there are goals that require you upgrade all your heroes to level 15. Or have 5000 bugs killed.
The Heroes section has awards based on each hero. For example, Beetle has an award that requires you swipe 10 bugs off a ledge in one swing, something only he can do. Each hero, however, has the same four awards: reach level 20, kill 2000 bugs, gain 100,000 experience and reach level 20 on coliseum. Only the three original heroes have unique achievements after that. Every other hero, whether it be a free promotional one or one bought from DLC, will only have the same four awards, which seems lazy.
The Scrolls section contains all the History Scrolls you've obtained throughout your adventure. It doesn't have a lot to do with awards however so it seems a bit misplaced.
Finally, there is the Game Centre section, which shows your other awards, with the signature green background, along with the points they add to your profile. All the awards add up to 1000 points when fully obtained.