The Frankenjura is our home crag, so we climb here more than any other place in Europe and know quite a bit about the area.
Why - Germany is a very American friendly, clean, beautiful country with tons of castles, nice cities, and delicious beer. The Frankenjura is also the birthplace of sport climbing, where "red points" were created and there are more sport routes here than any other place I have ever been. Thousands and thousands of quality world class limestone routes just waiting for you (over 8000 sport routes and 1000 crags).
Where - The Frankenjura covers a large portion of north eastern Bavaria. If I were coming for a road trip, I would station myself close to Pottenstein, Neuhaus a.d. Pegnitz, or if you want to be in a larger city, perhaps Bayreuth. You'll be 30 minutes or an hour from Nuremburg and about 2 hours north of Munich. For flights, NUE is the closest (1 hour), then MUC (2 hours), then FRA (3 hours). You will need a car. The cliffs are all scattered pretty far apart from each other.
Accommodations - Since we live here, I can't really recommend hotels. Frankenjura.com has a wealth of information though on the climbing here and even accommodations. Guesthouses can be as cheap as 30 euros a night around here and are quite nice. There are campgrounds, but if you have kids, since guesthouses are so cheap, it might be more economical to just do a guesthouse.
Food - Every town in the Frankenjura has great restaurants with pretty cheap food and beer. They tend to open at 1700 for dinner. Not as many places are open for breakfast and lunch, but you will find a few cafes and lots of bakeries. Every town has a bakery. A great restaurant in Neuhaus I recommend is the castle on the top of the hill, Veldenstein Castle. The castle is also a hotel and is really great for kids. They have a playground below the outdoor beer garden and amazing views/exploring all around the grounds (for free in the evenings). Get the Knights Plate. You won't regret it.
Crags - Wow. This is a hard area to cover because it really depends on where you decide to stay. There are SO many wonderful crags. As for going with the kids, these are some of the places we find to be the easiest:
1) Weissenstein (roadside crag) right by Neuhaus
3) Maxmillian Forest (lots of great crags in these woods)
I could probably list about 20 more that you can even bring strollers to, but check out the guidebooks. They show which ones are kid friendly.
When - Germany has pretty mild summers. I would say anywhere from May to October would be the best time for climbing in the Frankenjura. Spring can be pretty wet, and fall can be pretty cold, so summers are really ideal. Plenty of the crags are shady for the rare hot weather we do get.
What to do in the Area: Go to the Frankische Wunderland with the kiddos! My son loves it, and I do too! It is located right by Plech (a climbing town). Super fun amusement park that is real old fashioned. Go to an indoor/outdoor pool! There are nice ones in almost every decent sized town. The one in Hersbruck is particularly nice and right in the middle of a ton of climbing areas. Visit the big cities (Munich and Nuremberg).
If you need any more info about the area, shoot be a comment and I'll see what I can do to help you.