These chords are called secondary chords, and they're read (in this case) as "IV of IV."
The concept is that the chord is best understood in relation to another chord which isn't tonic. In this case, you're in the key of B♭. "IV/IV" means that it is the IV chord of the IV of B♭. In other words, it's IV of E♭, and IV/E♭ is A♭.
As for why it doesn't sound of out key, that's partially opinion. Some will think it does, some will think it doesn't. But this IV/IV (or ♭VII) is a common feature in popular music. We can conceptualize it as part of the Mixolydian mode, or borrowed from the parallel minor, but it's so common that you might not think of it as being chromatic.
And, if I may say so, typically Roman numerals are best used to show how a chord is functioning. Not knowing the rest of the piece, I see no reason to label this chord as IV/IV; it could just as easily be labeled as ♭VII.