I am doing a bachelor project in Algebraic K-theory, and have run into the swallowing lemma, in particular in the proof that the $Q$ and $S.$ construction yield the same and in the proof of the Waldhausen localization or fibration theorem. I was able to prove this result, but I am very uncomfortable actually using this result, or having intution for what it "really" says beyond being a neat coincidence of the theory.
Usually my knee jerk reaction to this kind of confusion is to wander around wikipedia, the nLab or more generally any corner of the internet, to get a feel and perhaps place this result in a bigger picture. However I am having a lot of trouble doing this with the swallowing lemma. Hence if anyone has any ressource of any kind or intution to help me "really" understand the swallowing lemma, that would very helpful.
Just in case, here is the statement. Let $\mathcal{A}$ be a subcategory with all objects of a category $\mathcal{B}$. There is a bicategory inclusion $\mathcal{B}\to\mathcal{AB}$. The former bicategory is the bicategory which is constant equal to $\mathcal{B}$ and the latter is the category whose horizontal arrows are in $\mathcal{B}$ and whose vertical arrows are in $\mathcal{A}$. This inclusion is a homotopy equivalence after geometric realization.