For me, sitting with a feeling is to do as already stated - to watch it. It is to allow it to exist without me trying to force it into being something else or trying to make it vanish. It is also about being able to make my actions not a response to the emotion. If I am angry, I can honor that anger by just hearing it - by letting it rant and rave - but without me slamming the door in response to it.

Sitting with my feelings is like my T sitting with me.

T can watch and witness everything I bring into session.
T can even be moved by what I bring into session, he may shed a tear or laugh.
But T remains T. T does not witness my stuff and then allow himself to get angry or upset. He honors me and does not try to silence me.