noun
kitchen area; family circumstances
A somewhat dated or literary term referring to the kitchen and its immediate surroundings, or by extension, the domestic or financial situation of a household. The second gloss 'family circumstances' is a figurative extension of the kitchen-as-heart-of-the-home idea.
勝手元が広い家は料理がしやすい。
A house with a spacious kitchen area is easy to cook in.
あの家は勝手元が苦しいらしい。
That family seems to be in difficult financial circumstances.
勝手 can mean 'kitchen' in older usage, but now more commonly means 'one's own convenience' or 'selfishness'. 勝手元 is a compound that retains the older kitchen-related meaning.
Compound of 勝手 (katte, 'kitchen' in older usage) + 元 (moto, 'origin' or 'vicinity'). The exact historical derivation is uncertain, but the word likely referred to the area around the kitchen and later extended to the household's domestic situation.