Why do downloads in Steam stop when I begin playing a game?
Steam pauses downloads upon launching a game in order to prevent interference with connection quality that the game may need.
If you wish to bypass this feature, press Alt+Tab (or Command+Tab for Mac users) to minimize the game and return to the Steam client.
Under the Library tab, select Downloads from the drop-down menu and click the "Resume All" button. This will have all downloads continue downloading.
[NOTE: If you wish to only continue an individual download, you may click the Resume button for that title located in the bottom right-hand corner of the title's download bar.]