This is the 11491st most frequent Russian word.
домашнем
"Домашнем" is the prepositional case form of "домашний," meaning "domestic" or "home-related."
In this sentence, 'домашнем' is used as an adjective in its prepositional case, agreeing with 'уюте' to mean 'home' or 'domestic' as an attribute of the noun.
Я нахожусь в домашнем уюте своего дома.
I am in the cozy atmosphere of my home.
Here, 'домашнем' is used to describe the noun 'ужине' (dinner) and implies that the dinner is homemade or familial.
На домашнем ужине мы вспоминали детство.
At the home dinner, we reminisced about childhood.
'Домашнем' here modifies 'матче' (match) to indicate that the match took place on the home team's field or territory.