We take it for granted these days that the stars are unimaginably bigger than the planets and the moons. But when you look at the sky, it does not appear this way. The moon looks bigger and brighter than the stars. This leads me to wonder when astronomy first discovered the relative size of the stars. Did people believe that the moon was bigger than the stars for a long time? If so, where was this believed, and up until when? What discoveries or observations led people to conclude that the stars are larger?
It was known since antiquity that stars have no parallax visible by naked eye (parallax due to the displacement of the observer on the Earth). This indicates that they must be much further than the Moon (Moon has significant parallax which was known in antiquity).
The next step was heliocentric system of Copernicus. If one accepts the motion of the Earth around the Sun, the stars must have parallax due to this motion, unless they are VE-ERY far. Actually this was used as an important argument against the heliocentric system. The argument was that they cannot be THAT far. The universe cannot consist of mostly empty space. Why would God create such a strange universe?
As they are very far, it follows that they are very much BRIGHTER than the Moon and some of them have to be brighter then the Sun, because we see them.
Strictly speaking this does not imply that they are bigger (they can be brighter but smaller, in principle). But if we make a natural assumption that they are of similar nature to the Sun, then their brightness and distance indicate they they are much bigger than the Moon, and some of them are bigger than the Sun.
All this could be made precise when the parallax (due to the Earth motion around the Sun) was eventually discovered (in 18th century), and when it was proved that the nature of the stars is indeed similar to the nature of the Sun (by spectral analysis, in 19th century).
Direct measurement of the size is still impossible (except perhaps of the closest ones), they all look like dots, even with the most powerful telescope. But we have a theory about their inner structure, this theory is well confirmed and it predicts their actual dimensions.