“Relax," said the night man,
"We are programmed to receive
You can check out any time you like
But you can never leave!"
Logically, after you check-out of a hotel you've surrendered your right to abode at that location - after that you're usually limited to common/shared areas like the lobby, bar, restaurant, maybe the pool - but excepting the lobby those places are closed at night - and they'd have security to remove people from the lobby if necessary - so as far as the Eagles' are concerned, what is it to "never leave" when you legally cannot stay?
> The song has been described as being "all about American decadence and burnout, too much money, corruption, drugs and arrogance; too little humility and heart." It has also been interpreted as an allegory about hedonism, self-destruction, and greed in the music industry of the late 1970s. Henley called it "our interpretation of the high life in Los Angeles", and later said: "It's basically a song about the dark underbelly of the American dream and about excess in America, which is something we knew a lot about."
Hotel California is, of course, not literally a hotel; it's a metaphor for an addictive and entrapping lifestyle, and your legal "right to abode at that location" is a real-world detail that doesn't really matter for the purposes of the metaphor. The singer wants to get out -- by "checking out" he has declared his intentions to leave the hotel, but the point of the song is that wanting to leave is not the same as actually leaving.
It's a bit more obvious if you consider the full verse:
> Mirrors on the ceiling
/ The pink champagne on ice
/ And she said: "We are all just prisoners here
/ Of our own device"
> And in the master's chambers
/ They gathered for the feast
/ They stab it with their steely knives
/ But they just can't kill the beast
> Last thing I remember, I was
/ Running for the door
/ I had to find the passage back
/ To the place I was before
> "Relax," said the night man
/ "We are programmed to receive
/ You can check out any time you like
/ But you can never leave!"