First thing I typically do when that happens is update my system and reboot. This is useful for ensuring everything is in a known consistent state and there is no weird runtime issues that happened since you last booted. And it is always good to upgrade before you reboot to ensure you are booting the latest kernel and drivers.

If that does not help then I would start by closing down steam completely (ensure it is not running in the systray at all). Then launch steam though a terminal and start the game as you normally would. You will hopefully see some logs for the game in the terminal. Though it is very game dependent as to if that will be useful at all. If not I would look online to see if the game logs anything to any other file as some games tend to do their own logging or have a flag you can enable.

If the game gives you some logs and hopefully an error message you can then see if it is useful to you and if not try googling for that error and the game name. I find this tends to dig up more specific help for games then general searches for terms like wont start or crashes though sometimes those general terms can find a solution as well.

Note: if you try to launch steam in the terminal and it is already running you wont get any logs at all from it - it basically just forwards things to the main instance or quits as it does not need to do anything. Only the first instance you start will give you and useful logs.