In general, when you have two Roman Numerals on top of each other they are known as secondary chords. They are chords that have function outside of the current key. The most common of these are secondary dominants which V7/V correlates to temporarily tonicizing V via raising the 3rd of ii7.
To this specific instance of iii/IV, the author is saying that they interpret this chord as functioning more in the context of IV than in the current key even though it can be looked at as vi. I don't necessarily agree with this and I would call this a non function secondary chord since it doesn't add much function to the analysis. Depending on the line I may even be inclined to call it
IV7 - iii - vi - V7 if the progression sounds more like vi is apart of the previous chord than an independent chord change.
In this progression, I'd be more inclined to talk about secondary chords in terms of vi since the iii goes right back to vi so someone could talk about the iii being used as a minor dominant hence
IV - vi - v/vi - vi - V7. While not being very strong, it would show the relationship better to what is going on. I'd honestly even call that a stretch.
If you are interested in learning more about non functioning secondary chords specifically non function secondary dominants, see this question.