Ah, okay, there isn't any easy answer/number/table for this.
In general, the "Aufsichtspflicht" results from Bürgerliches Gesetzbuch (BGB)
§ 1631 Inhalt und Grenzen der Personensorge.
There are not clear cut lines or given ages. This is mainly because it does not make a lot of sense. A very independent and mature 6 year old can do things a less developed or even disabled person might not be able to do, even at the age of 18+.
Generally speaking, the parents have the duty to care for the child, anything else can be handled by contracts. For example a contract with a kindergarten. Or School. Or private/public pool.
From personal experience I can say this: Germany is very safe. Kids go to school on foot, on bike or longer distances by normal public transport among all the other adults. We do not have "school busses". Between 07:00 and 08:00 public transport is packed with kids of all ages going to school. Schools do not normally have their own stops. Kids get off at the nearest stop and walk the rest of the way. Around the school you will see streams of children coming from all directions on bike or foot. It is rare to see adults accompanying them, unless it's the first weeks of elementary school or the last day before school holidays where parents pick up their kids to do something fun after school.
When I was a kid, I walked to school every day, almost from day one of elementary school. The first few weeks an adult would accompany us (one of the parents, they had a rotation going), then we walked on our own, as a group of 3-4 kids in the same class from the same neighborhood block. Google maps says it's a 10 minute walk of 800 meters. If any teacher had stopped me from leaving the school building on my own when school was finished, I am sure people would have wondered why. That only happened for mentally handicapped kids. Even kids who needed transport for one reason or another would just know to wait in front of the school. No teacher or other adult would be there.
At 5th grade, I know half my class came by bike and in the winter or on rainy days by public transport, because they did not live in walking distance to the school. Neither did I, but I lived close enough to not get the public transport ticket subsidized, so the kids from our street got there by bike, or on foot when the parents deemed riding a bike to unsafe in snow and ice in winter.
Who is responsible for the child now that the class has ended?
The parents. Unless the contract says otherwise, or circumstances are exceptionally dangerous. There is no exact written guideline for this, because it would not make sense. The adult has to know what is exceptionally dangerous and what is not. It can be perfectly safe for a 10 year old to walk home through the woods. If that is what the parents deem okay, then it is okay. Yet as the adult in charge, you have the duty to keep them back, if you know the woods are on fire today, or a criminal escaped into the woods or something similar that would make it significantly more dangerous then the parents had in mind when they made their general decision.
If a 10-year-old child says that their parents "said they could go home alone", can I reasonably stop them?
If you have a contract that says they cannot go home alone, sure. If there is a special situation that implies the conditions have changed from what your contract said (lets say the radio broadcasts a storm warning and you see that the weather is horrible and branches flying around could injure a smaller kid) you can (maybe even have to), even if you do not have a contract.
Otherwise... why would you? It’s their parents decision. If they had wanted you to keep an eye on their kids when the time is up, they would have made a contract that says so.