This quarry is still active, but it has been open to rock climbing for many years. Of course, not in the active areas where they are still using explosives. Which is something you must be aware of when visiting this area. Please only climb in the areas where climbing is permitted and do not wander. Please also don't litter, start new fire pits (there are three pits where fires are permitted) or harm the trees. Please do not camp in the quarry or at the parking area.
The rock is porphyry, which is kind of glassy and crystaline. There are a many cracks and many routes which are trad protected. The rock requires a good sense of body positioning and demands good technique over strength.
The town of Löbejün is small, but you can find much of what you need during your visit here including groceries (EDEKA, for example), restaurants/bakeries, fuel, and a hotel (Pension Renate).

There is indeed a way to get her via train, but you'll have to research it yourself. I imagine it's much more complicated than with a car, and it's probably not cheaper, either.
Löbejün is located 20 km north of Halle and has its own exit off the Autobahn A14. Follow the signs to reach the town, then take a right onto "An der Stadtmauer" and follow it for around 600m where a small road branches off to the right in a left curve. Turn right here onto this small road and follow it to the parking lot (the GPS coordinates indicate this parking lot).