When viewing star maps in Stellarium, I noticed that the meridians of the equatorial grid were denoted in hours and not degrees, and that always puzzled me. Why did astronomers choose this system? I understand it has something to do with being relative to the Earth's rotation which takes 24 hours, but I can't figure out exactly what.
In order to know when a star will be above horizon, you'll need an equation with times, not "celestial latitudes".
So you end up adding RA and sidereal time.
In order to find out the position of a star you need to find out how its Hour Angle varies with time.
The Hour Angle is how far the star is from the observer's meridian:
In this image, it is the angle between the site's celestial meridian (North Celestial P ole-Z enith-S outh horizon) and the star's celestial meridian (P-X-Y). It is the same angle as between where X culminates and X.
So it comes than $HA=ST-RA$ and since you may not have a "local sidereal time" clock at hand, you can use Greenwich's Sidereal Time: $HA=GST-Lat-RA$
Astronomers use this system simply because it is (was) convenient. The sky shifts about 15° every hour or a full circle, 360° every 24 hours (actually 23h56m). So if you want to know where a star is in an hour, simply add an hour to the local hour angle.