We find lots of question regarding Multi file upload coming grayed out in SharePoint.
After surfing through net, I found the working solution
When opening SharePoint document library, you would be asked to run Microsoft Office 2010 component by Internet Explorer automatically.
Means the issue belongs to MS Office not with SharePoint Directly. After going through above link
I found that there is an ActiveX Control named as STSUpld.UploadCtl in the Manage Add-ons optionmust be enabled in IE.
It belongs to the location
C:\Program Files (x86)\Microsoft Office\Office15\STSUPLD.DLL – For Office 2013
C:\Program Files (x86)\Microsoft Office\Office14\STSUPLD.DLL – For Office 2010.
You can found more information here also.
If the above missing you need to download the dll from here or need to enable the Add on if already exists as shown above.
For more information about STSUpld.dll is available on msdn article