The OED entry for "studio flat", a synonym that inspired the Americanism "studio apartment", says that the term originally (late 19th century, perhaps earlier) referred to an apartment which would be suitable for use as an artist's studio. The term probably entered real-estate jargon from there. One of the attestations included, in an architectural publication from the late 1940's, especially reinforces this etymology:

At the foot of the San Jacinto mountains, this small group of studio apartments is keyed to the needs..of a specific class of tenant—artists who spend part of the year in Palm Springs.

By 1970 it seems that the term had become applicable to any sort of one-room dwelling (this quotation from a novel):

He had what they call a studio flat—bed, gas fire and tiny kitchenette.