Turning on labels of layer using ArcPy?

I think your problem is that your code is enabling the checkbox under the Layer Properties that says "Label features in this class". The part you are missing is the code to enable to the checkbox for "Label features in this layer"

Try inserting this code:

layer.showLabels = True

After your if statement that activates the label classes, like the following:

import arcpy
mxd = arcpy.mapping.MapDocument(r"Mypathhere") #Map document reference
layer = arcpy.mapping.ListLayers(mxd, "Layername")[0] #Indexing list for 1st layer
if layer.supports("LABELCLASSES"):
    for lblclass in layer.labelClasses:
        lblclass.showClassLabels = True
layer.showLabels = True`
arcpy.RefreshActiveView()
mxd.save()
del mxd

If you are not always using label classes, and simply want to turn labels on, then the code can be simplified to:

import arcpy
mxd = arcpy.mapping.MapDocument(r"MyPathHere") 
layer = arcpy.mapping.ListLayers(mxd, "LayerName")[0] 
layer.showLabels = True

and if you want to do things like refresh the data/layout view (whichever is active), see the checkbox marked in the Table of Contents, and/or to save the changes to your map, then just add lines like:

arcpy.RefreshActiveView()
arcpy.RefreshTOC()
mxd.save()