I found this on the nLab (at the start of the "Top-, Kan- and simplicially enriched categories" section):
"... an $(\infty, 1)$-category is supposed to be like an enriched category which is enriched over the category of $\infty$-groupoids. This turns out to make sense literally if one takes care to remember that $\infty$-groupoids themselves form a higher category."
I don't feel that the last sentence is explained in the article. Does anybody know of a reference where this is fleshed out?
Enriched $\infty$-category theory has been developed for instance in Gepner and Haugseng's 2015 article Enriched $\infty$-categories via non-symmetric $\infty$-operads. It is Theorem 5.4.6 that the $\infty$-category of (small) $\infty$-categories is equivalent to the $\infty$-category of categories enriched over the $\infty$-category of $\infty$-groupoids. It is admittedly not an easy read, however.