I agree it is a bit confusing. Layer files (LYR) are currently not shown in ArcMap because they don't actually contain any data. Thus, the likelihood of them working by someone other than the person who created it is small. That being said, you can open the layer file from the arcgis.com website by clicking the Open link, which will launch ArcMap.
I suggest you create a layer package and upload that instead. Then you will be able to see and access it directly from ArcMap. Also, anyone who can see the layer package will be able to open it.
Hope this helps,
Mike