In Overwatch, there is a meter at the bottom left corner that indicates how "on fire" (as in, doing well, I'm guessing) you are. When it get over an indicated point, your portrait on the tab screen/scoreboard is shown with a fiery background.
Is there an actual gameplay benefit to being on fire?
To quote the Wiki,
"The on-fire meter is an indication to the player as to how well they are performing with a hero. The meter starts off as blue, and turns to red as they carry out actions that adds to the meter. When "on fire," the hero's character portrait gets a flaming halo. The meter can get more 'fiery' through actions such as scoring kills, completing objectives, and denying objectives to the other team. The enemy team will also know if a hero is "on fire" as a mechanic to alert other players of enemy player skill, allowing them to adjust their tactics".
On-fire status is also shown on the team summary screen shown during matches.
That said, it's basically just an indicator of skill, displaying how much of a threat you are to the opposing team, which may be useful if you want to cripple the enemy team by killing their MVP.
It pretty much only means that you are "On fire"/"Doing very well". Your portrait is also on fire when it reached the upper limit.
Looking at the scoreboard, you can also see how many of your teammates/enemies are on fire.
Think of it as (If you play Team Fortress 2) that player has a high kill-streak and is one of the more dangerous people on the team.
Currently it doesn't seem to have any effect on the game (Exp, loot, etc), but the developers may change that.
