For Windows 7, please make sure you have Service Pack 1 installed. That is the minimum required version of Windows.
Check for updated graphics drivers and Windows updates in general.

If you are playing on a laptop with dual graphics, make sure the game is loading on the discrete graphics (nvidia or AMD) rather than integrated (Intel, though HD 4000 and later should have DiretcX 11 support).

Make sure your graphics card has hardware support for DirectX 11. One way to check which card you have is to click Start (or WinKey-R), type in dxdiag and hit Enter, then in the window that opens, switch to the Display tab.
Also in that tab, check the 'DDI Version' in the Drivers section on the right; it should be 11 or higher. If it is less than 11, still check the video card specs, since there are driver issues and some programs that can result in the DDI Version being lower than what it should be. In that case, a clean re-install of the drivers may help. If possible, check your monitor/laptop manufacturer's site for an updated inf file, especially if a monitor is just listed in Windows as a generic PNP monitor (that is often the case for laptops).
It is also possible for the DDI Version to be listed higher than the actual hardware support. For example, Intel's integrated graphics drivers can cover an entire family of GPUs and will report one number, even if they include chips with different levels of DirectX support. Intel HD 2000 and 3000, for example, only support 10.1.

Try manually installing the support packages (from the folder ..\Divinity Original Sin Enhanced Edition\__redist) and reboot.

Did you check for updated graphics and monitor drivers?

Are you shutting down all non-essential programs before trying to start the game, especially anything graphics related? (video recording, graphics monitoring/tweaking, anything with an onscreen display, etc)

Please email supportdos@larian.com, with the report.zip file generated by the D:OS EE Analysis Tool (or post/PM/email a Dropbox or Google drive link, etc).
Extract the zip file somewhere and run AnalysisTool.exe. When that loads, browse to the install folder (if it is not automatically detected), optionally run the tests first, and then generate a report (it will be saved in the same location as the Analysis Tool).