Windows wants to know what program to use to open/install LibreOffice.
If double-clicking on it doesn’t automatically start the install process, the file is probably damaged, or you changed the extension so that Windows doesn’t know the file type anymore.
EDIT: It works for me (LO 5.2.1.2 32 bits), maybe there’s a corruption in the 64 bits version. If downloading again doesn’t solve it, you can file a bug here: https://bugs.documentfoundation.org//
EDIT (15-9-2016, for Lupp) Just installed LibreOffice_5.2.1_Win_x64.msi without any problems on Windows 10. So the file is fine, if you have trouble with it, something must have gone wrong downloading or saving it.
(Off topic!) You added “9/15” as a date to your edit above. Why do you use the (undue abbreviated) USA date format in this international forum? It is only common in the US (and probably in Belize and in parts of Canada) and it is causing trouble again and again. From a systematic point of view it is the worst format I know (except ancient Roman).
Okay, edited. Actually it’s not that bad when the day number exceeds 12 so you can’t misread it. 
Msi installation file is broken.
After validation, there is a lots of errors in ICE03. It seems to bad language value in Files table.
Windows 10, and GPO can not allow to install from broken package.
Tested versions:
LibreOffice_5.2.1_Win_x86.msi
LibreOffice_5.2.1_Win_x64.msi
LibreOffice_5.2.2.1_Win_x86.msi
LibreOffice_5.2.2.1_Win_x64.msi
Last known working package is 4.3.5.
I am having the same issue. I never changed anything that I am aware of. The first time that I attempted an update after “upgrading” to Windows 10, windows wanted to know what program I wanted to open the update file with - and I can’t ID a program that works.
How does one resolve this issue???