The "splash" screen will be about 1/4 of your monitor's space, but the game itself should open full-screen after going through the loading preliminaries. With a Steam game, it is best to start the game from the Steam menu, whether you start the Steam app itself and use the menu there, or you use the Windows Start menu and start it there. If you start it from the SH5.exe file in the game folder, it will attempt to start through UPlay, which does not function correctly in a Steam install. Steam wants to do its license check on your game...
What is the "native" resolution of your display monitor? Windows will show that as "recommended" in the display properties for screen resolution. You want to set Windows to that resolution. Then once inside the game, on the main menu is an Options choiice, and on it, a Graphic Settings tab where the game's resolution can be set, if it's not correct... If your game is minimizing to the Task Bar and not "opening" all of the way, then the questions are "did you install the DirectX v9.0c and Media Codecs from the game install menu?" and "which folder is your game installed in on your hard drive?"
