Sometimes games, especially Steam games don't start up but their Windows process remain active.
You can check by:
1. Start the Task Manager (If your on Windows Vista/7 simply type :"Task Manager" into the Start Menu Text box or just press CTRL+ALT+DEL and select Task Manager from the list)
2. Go under the "Processes" tab
3. Keep the Task Manager open and then start the game
4. When the game closes after the brink picture, check the Task Manager and see if there is still a process called Brink.exe running there.
If there still is a process there then the solution can be very simple sometimes, even trivial. Try to restart your computer and try again.
I know how stupid this might seem to you, but this is the only thing that I can recommend at this points since you don't really offer enough information to present more valid theories.
For example,
Describe your system including to which version of Windows are you using.
Some anti-virus software asks you to allow a program to access your computer when you execute that program for the first time. Perhaps this was in your case and you accidentally denied the program the access.