I get that an exchange can use a UIA in BTS's DEX, but what is the mechanism that they tie that UIA to something of value?
The value of UIAs is based on trust in the issuer. UIA holders are exposed to counterparty risk with respect to the issuer who is backing their value.
The part that is not clear to me is where is the collateral stored? How do we know that the UIA can be redeemed?
The collateral/reserve in the case of the UIA is stored off of the BitShares blockchain. If this is a UIA that is an IOU for some cryptocurrency, then the reserve will likely be stored in that cryptocurrencies' native blockchain (likely controlled through keys held by the UIA issuer). If the UIA is an IOU for fiat, then the reserve will likely be fiat held in their traditional bank accounts. Note that they do not have to have full 100% reserve backing their IOUs but customers should demand that they do (also it should at least be more transparent to check on the blockchains in the case of cryptocurrency IOUs). The UIA may also have no real reserves and just be a promise to pay back the UIA holders in the future with something of value. Again, you are trusting the UIA issuer who is backing the value of the UIA and there is no guarantee
that you can redeem the full value promised. You just have to trust that they actually will pay it back to maintain their reputation and continue their business model (and also their jurisdiction will likely consider it a security and therefore they will have a legal obligation to make their customer's whole if possible).
The UIAs are interesting because we want gateways to use them as a way of getting fiat into and out of the BitShares system. The companies we would typically be targeting to act as gateways are exchanges that have already enabled bank transfers of fiat into and out of their existing system. These exchanges are used to making their money through percentage fees on each trade, so their revenue scales with the volume of trading of their assets. We are asking them to not focus on the exchange features (order books, matching bids and asks) and instead only focus on the fiat gateway portion. Some exchanges may not be comfortable with such a huge change in their business model. This new feature now allows these exchanges to use their same business model to make money except BitShares does all their order book work for them and also stores the cryptocurrency assets so that liability isn't on the exchange anymore.