Here's a visual representation of the scenario
Here you can see we have a black hole on the left. The event horizon is the edge of the black hole. You are far away from the event horizon, and a chain is passing you by fast heading toward the black hole. Due to the size of the black hole and the makeup of the chain, the chain will not break before reaching the event horizon.
After a while, the following scene happens:
As the end of the chain approaches the event horizon, the chain slows down due to the immense gravity as it approaches a frozen state.
As I show in the diagram, you can observe this phenomenon as well as observe the chain moving fast by you toward the black hole.
How can the part of the chain near you appear to be moving quickly toward the black hole, while the end near the black hole is frozen (or close to it)? Where does all that chain go?
Let me ask the same question in another way..
If the distance between you and the black hole is 1000 units, and the chain appears to be almost frozen 1000 units away from you, how could you reconcile watching 10000 units of chain speed past you? How does that 10000 units appear to fit within a distance of 1000 units from your perspective?