Let $k$ be a field.

A folklore theorem states that dg-categories (over $k$), $A_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example Stable infinity categories vs dg-categories).

I would like to understand in what sense (and why) dg- and $A_{\infty}$-categories are "the same".

There is a canonical inclusion functor from the category of dg categories to the category of $A_{\infty}$ categories. However, this functor is not full, so it does not define an equivalence of categories.