The bitcoins that are on the exchnages are usually off the chain transactions ie you send the money and your account gets credited, but those coins will go somewhere else like a hot or cold wallet so you essentially just have an IOU.
That sounds like a recipe for disaster - what measures were put in place to avoid fraud? If any? That is to say - why did so many people feel comfortable trading with/within such an entity?
Many people felt comfortable leaviing bitcoins in their Mt Gox accounts because they were unaware of all the troubles that Mt Gox was having.
Anyway, the practice is common. For example, you don't really own any of the shares in your stock broker account. They are owned by the broker.
It is also necessary. Exchanges could not function efficiently if trades went through the block chain.
