If you receive an MMS message in the Messages app on your Samsung Galaxy S5, then it can happen that you tap on the download button, but the download is not performed. This usually has the following simple reason:
If you try to download the MMS with a WiFi connection, then this will not work. For being able to download a MMS message on the Samsung Galaxy S5, will need to authorize yourself via the mobile data network. Only then the MMS message can be downloaded.
So activate the mobile data connection on your Samsung Galaxy S5, to download the MMS message. Thats it!
You should now be able to download it completly and to read the MMS message.