I've been playing along at home, and have a partial explanation of what's happening.
On the left are the tracks in an album, displayed in the correct track number order. On the right is the actual order in which they are playing. In this case they are playing backwards, but in other albums, the playing sort order (on the right) appears to be random.
2. Perhaps you have (accidentally) activated shuffle play or Auto DJ?
My money is on this.
The thread title and description isn't quite accurate, and may have sent some people down the wrong path.
MusicBee isn't
playing tracks out of order - it's
queueing them out of order.
Even with shuffle enabled, queueing an album will usually still list the tracks in album order. It's only during playback that the order is randomised.
(That's why the
Play Shuffled command exists - it will actually queue tracks in a random order)
As for why it's queuing them out of order, that's trickier and I think is being hidden by your particular
Now Playing panel settings which I'm struggling to replicate.
(It's not clear if that's your entire window, or just part of it)
Consider these two views of the same album:
This an album selected in the
Library Explorer on the left, with the tracks from the album shown in the middle:

Here the tracks are shown in album order.
Double-clicking the album title (or
right-click > Play Now) will queue the tracks in the same order.
Now - with the same album selected - I have sorted the tracks in the middle tracklist by title:

Playing the album now will also queue the tracks alphabetically by title.
Even though the tracklist isn't shown in your view, somehow I think you have accidentally sorted the tracks by a field other than track number and MusicBee is queueing tracks in that order.
It might be a bug, or it might just be an unfortunate side effect of your particular panel configuration.