The IV/IV would be called a "borrowed" chord. It's never followed by IV, so it is not technically functioning as a secondary subdominant. It is borrowing notes from another scale. We generically call this type of pre-dominant chord (precedes the Dominant) in a cadence a "substitute subdominant".

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I agree with all the other comments. One thing I'll add is that, while this doesn't make sense in this context, secondary chords are very useful in cases like the "Hey Joe" progression (bVII IV I) or circle progressions (basically extended II V I sequences)

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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♭.

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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.

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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.

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It is the Sub Dominant chord of the Sub Dominant Key. So if the Tonic key is Bb then it is the Sub Dominant chord of Eb Major, so Ab / C / Eb. It is a modulation.

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The chord doesn't belong to the original key. From what I understand, this is an "applied chord". What does that mean? Why doesn't it sound out of key?

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