This is an idiom, it means something like ‘happily, well’. Maybe it’s because cheese tastes well with butter. Here’s an explanation:
like cheese in butter
Usually when people are extremely satisfied with their workplace and quality of life, they feel very comfortable, lucky to be well-paid, and appreciated. This expression is used in such situations.
This explanation was taken from an article about Russian, but this idiom is shared between Ukrainian and Russian. Polish also has something similar, but with cheese replaced by a donut (apparently Polish people prefer donuts to cheese).
Suggested translations are ‘(to be) comfortably off, (to live) in clover, (to live) off the fat of the land, (to live) like a king’.