The vertical arch runs from due North to due South, thus dividing the sky into Eastern and Western hemispheres. The imaginary line in the sky it traces is called the Local Meridian. This can be used as a crude sort of timepiece- the time that the Sun crosses it is your local solar noon. The particular time that other objects cross that arch is called the transit of that object, and it marks the moment when that object is highest in the sky.
If you combine accurate timekeeping with two of this kind of observatory, you can use the relative time difference between the transit times recorded between the two to figure out the difference in longitude between them. The modern system of geocoordinates more or less assumes that such an observatory exists [EDIT: ...at Greenwich, England. Omitted that before.] This is the same principle as a sextant.
At a single observatory, you can also use the time between the transits of two different objects in order to determine the East-West angular separation between them. If you use the standard reference of the location of the Sun on the Vernal Equinox, this gives you the standard sky coordinate called Right Ascension.
The slanted arch marks the line on the sky of points that would be directly overhead for people standing on the equator- the Celestial Equator. The angular separation between this arch and some object in the sky is the standard sky coordinate called Declination.
I'm sure there are other uses for it, but those are the major ones that I could think of.