The song is about a man who's talking by telephone to the woman he loves, when he realizes that another man is with her.
Put your sweet lips a little closer to the phone
Let's pretend that we're together all alone
I'll tell the man to turn the jukebox way down low
And you can tell your friend there with you, he'll have to go.
This is the story followed in the video (more or less).
Hugh
PS. I'm sure that when Jim Reeves sang this in 1959 the 'phone' would have been a landline phone and not a mobile!