Translation:Where is the kitchen?
Great question! I believe it's a coincidence, as 所 is actually usually pronounced ところ. It's only being pronounced as どころ here because it's part of a compound word (and a unique one at that, pronunciation-wise) and because of rendaku.
所 is also colloquially pronounced とこ and commonly pronounced しょ in more typical compound words, e.g. 場所 (ばしょ, which also means "place" though confusingly, it's somehow subtly different from plain old 所).
This is an interesting question. A proper linguist or historian could probably give you a more accurate answer, but, having looked into it briefly, it's my understanding that 台所 is a shortened version of the phrase 台盤の所, hence why on'yomi and kun'yomi are getting mixed up. Apparently,「台盤 (だいばん)」 refers to the low tables which were essentially trays with legs that were used to serve food and drink during the Heian period. When not in use, these tables were kept in the kitchen so that they could easily be loaded with food. Thus the kitchen became known as "the place with the low tables" or 台盤の所, which over time became 台所.