What exactly does this "play a country song backwards" joke mean?

Many country songs especially older ones talk about things that are lost - wife, family, freedom, truck, better times, etc. Playing the song backwards gets you everything back because you un-lose them by playing it backwards. You get everything back. Un-lose. It's funny to me. I don't think there's any intention to speak to backward masking with that joke. That was a rock music thing not a country thing.

Also the relationship between country music specifically is that those themes of loss are typical of country music while not so much for rock music or other popular music forms.

In the 1960's, some music groups recorded tracks on songs that sounded like gibberish, but, if the record was played backward (remember, these were vinyl records, so you could place the record on your turntable, and the needle on the record, and turn the record with your fingertip), the gibberish would become ungarbled, and a clear message could be heard. (This is called backmasking.) Perhaps the most famous of these messages was one that said "Paul is dead" on the Beatles' song Number 9. (This example is so popular, you can still buy a T-shirt.)

Country music is known for lyrics where songwriters lament about things that are lost (lost loves who have walked away, lost fortunes that have been squandered away, etc.). David Allan Coe even made fun of this, in a way, at the end of his song You Never Even Called Me by My Name:

(spoken) Well a friend of mine named Steve Goodman wrote that song
And he told me it was the perfect country and western song
I wrote him back a letter and I told him it was not the perfect country and western song
Because he hadn't said anything at all about momma or trains or trucks or prison or gettin' drunk

(still spoken) Well he sat down and wrote another verse to this song
And he sent it to me and after reading it I realized that my friend
had written the perfect country and western song
And I felt obliged to include it on this album
The last verse goes like this here

(sung) Well I was drunk the day my mom got out of prison
And I went to pick her up in the rain
But before I could get to the station in the pick-up truck
She got runned over by a damned old train

So, put those two things together – backmasking, and recurring themes in country music – and you should be able to understand the joke.

In country songs the subject is often woeful, at least stereotypically it is. Typically the songs lyrics revolve around losing things such as your house, your wife (or husband), your dog, your vehicle (typically a truck). If you were to play it backwards, as if time itself is rolling backwards, you stop losing all those things and gain them instead.

Country music tends to be rather melancholy, and a stereotypical song might be "I lost my truck, I lost my dog, I lost my wife, I lost my home."

Reversing its meaning gives you it all back again.

The joke is that you cannot reverse the meaning by playing the song backwards.