Are you familiar with the centralized exchanges like Poloniex or Bittrex?
In part, Openledger works in the same way. When you deposit BTC with Poloniex, it goes into some wallet that's controlled by Poloniex. They have a database which tells them how much of their BTC belongs to you, and then they basically give you an IOU that you can trade on the Poloniex exchanges.
With Openledger, you deposit BTC and it goes into their wallet. Just like how Poloniex has a database which tells them how much of the BTC belongs to you, Openledger also has such a database - but their database is the Bitshares blockchain itself. The way they track how much of the BTC belongs to you is that they issue you an equivalent number of OPEN.BTC tokens.
So Openledger keeps the BTC that you deposit, and I suspect it's auditable in some way but I've never tried.