Chord algorithm - how to map nodes to keys?
Chord implementations use a defined function which is used to map the key identifiers (0, 2, 8, 11, 15 in your case) to the node identifiers (1, 5, 7, 10, 13). In the example you provided, the function works like this:

NodeID = KeyID % N


where NodeID is the (identifier) node where the key will be inserted, KeyID is the identifier of the key to be inserted, and N is the total number of nodes.

After you calculate the NodeID, it might be the case that this NodeID does not exist. For example, when KeyID is 0, the NodeID will be 0. Since NodeID doesn't exist, it will be allocated to the next highest (existing) node identifier which is NodeID = 1.

The keys identifier can be calculated using some hash algorithm.