I have skimmed some of the documentation/white paper for Swarm, the decentralized storage system, but I'm having difficulty understanding how the incentive system works (i.e. motivation for people to let other store data on their nodes.) Can someone with a better understanding explain succinctly how this will work?
There are two main incentive systems within SWARM, Bandwidth Incentives, and Storage Incentives.
Bandwidth incentives use the SWAP (Swarm Accounting Protocol). Basically each node keeps sets a highest accepted chunk price (how much you are prepared to pay per chunk) and a offered chunk price (how much you are asking to deliver a chunk). When information is retrieved each node keeps a record of the balance of payments between it and each other peer. Peers can either deliver chunks or send checks/micropayments to each other to maintain the balance of payments. If a peers debt becomes too large it is disconnected and its reputation suffers.
Storage Incentives are more complex. The first incentive is the same as bandwidth incentives you are compensated for delivering a requested chunk. However that provides no incentive to store unpopular chunks long term. So the idea is that nodes that are willing to provide a long term storage service put up a security deposit. They can then sell storage services and SWEAR to store certain chunks. They can then be challenged (SWINDLE) and if they cannot prove that they are storing the chunk they promised they lose their security deposit.
For more detail see the orange paper