I have the code
string xsltPath = System.Web.HttpContext.Current.Server.MapPath(@\"App_Data\") + \"\\\\\" + TransformFileName
It returns
You need to specify that you want to start from the virtual root:
string xsltPath = Path.Combine(System.Web.HttpContext.Current.Server.MapPath(@"~/App_Data"), TransformFileName);
Additionally, it's better practice to use Path.Combine to combine paths rather than concatenate strings. Path.Combine will make sure you won't end up in a situation with double-path separators.
EDIT:
Can you define "absolute" and "relative" paths and how they compare to "physical" and "virtual" paths?
MSDN has a good explanation on relative, physical, and virtual paths. Take a look there.