It wasn't until invention of the telescope in 1608 and the curiosity of Galileo Galilei that things started to change. Galileo found that under magnification, the band of light separated out into thousands of individual stars.
Not much changed until the 18th century when another astronomer, William Herschel who was working from his own observatory in England, turned one of his large telescopes on the Milky Way to try and measure the distance to as many stars as possible.
Making the rather rash, yet incorrect assumption that all stars give off the same amount of light, he estimated their distance based on apparent brightness in the sky, fainter ones being further away than the brighter ones. We now know that stars vary considerably in the amount of light they give off, so his distance estimates would have been quite wrong even though he was just working on relative distances rather than absolute.
ANALYSIS: Discover Your Own Tiny Galaxy
That said, Herschel correctly drew the conclusion that we are located inside a giant disk of stars with the Milky Way representing the plane of the disk.