Liberty Reserve's main line of business was operating a shared ledger trusted by multiple companies which would exchange cash for ledger entries and ledger entries for cash, earning a fee each time. Think like Western Union except minus all the pesky regulated bits.
LR was a direct reaction to what had befallen e-gold, which was in fundamentally the same business except it did all of the cashing first-party (early in life, at any rate). e-gold was shut down for being a money laundering operation, principally because it was a money laundering operation.
LR was also a money laundering operation. It observed various issues e-gold had had, such as being physically located in the US (though theoretically registered in a country friendly to money laundering), and intentionally fixed these bugs with the intent of being the most robust money laundering operation it could be.
This ended up not working out, which is why the vacuum that urgently desires a money laundering operation has replicated the same hub-and-spokes model with the hub being a cryptocurrency (presently Bitcoin) and the spokes being exchanges and LocalBitcoins sellers. The onramps/offramps to Bitcoin are exposed to a variety of ways to curtail their operations but the central trusted DB required for them to cooperate is not controlled by anyone who can be conveniently extradited. People in the Bitcoin community refer to this obliquely as either "censorship resistance" or "resistance to policy attacks."
If you're interested in the history of this, Paypal and e-gold were approximately contemporaries, and one of the reasons Paypal succeeded and e-gold did not was that Paypal conspicuously courted respectability, so that even if Paypal was also doing hundreds of millions of dollars of fraud and illicit transactions it would be unlikely to indict them as a criminal enterprise because they were a proper company whose lawyers wore suits and whose VCs were quite respectable. Bitcoin collectively cottoned onto the wisdom of having its fronts benefit from the same factor as opposed to being, well, insufficiently socially insulated from legal process. (I can accurately summarize who makes Bitcoin exchanges but it would read like a parody.)