This is awarded for reaching the 25th Level on your Online profile.
In the top right corner of the main menu you'll see a number of icons:
The number beside your Gamertag is your Driver Level.
This number is coloured Bronze  when you start, and your goal is to get it to Bronze  as that is Level 25.
There are around 200 Driver Levels in total:
> Bronze  to Bronze 
> Silver  to Silver 
> Gold  to Gold 
> Platinum  to Platinum 
You increase this Driver Level by gaining XP in Online Multiplayer sessions. This can be done in:
> Weekly Events
The amount of XP you earn is based on a number of factors:
> The length of the races and the total distance you drive
> The number of players in a Lobby
> Your finishing position
You can earn XP by pausing the game and letting the AI drive for you instead, however the totals are much lower than if you drove manually.
From my experience, a 5 Lap Ranked Multiplayer race would give you between 3,000 and 5,000 XP depending on how many players are in the lobby and how well you did. Qualifying would yield between 500 and 1,000.
An AI-driven 50% Unranked race yielded about 7,000 XP.
Driver Level 50 is earned when you have roughly 1,130,000 XP in total, which puts Driver Level 25 at around 400,000 XP since the XP required to reach each successive Level increases as you go along.
You will likely unlock this naturally as you go through the Multiplayer side of the game, so I recommend you avoid focusing on it until you complete the other Achievements in this area as you'll naturally gain XP from those.
If you continue along your way you'll likely earn this: