Well, the basic premise was as follows. Not all of it is based in fact, and in reality some of it was just flat-out bullshit.
Hundreds if not thousands of illegal weapons transfers happen every year. The main threat is folks buying guns in the US and then shipping them to Mexico illegally to be used by criminal organizations like the cartels. They're difficult to track because by the time you realize its happened, the guns are gone.
So, if you start by allowing something illegal to happen, such as an obvious "straw-man" buyer (that is, someone buying the guns for another person who otherwise can't legally buy their own guns), rather than trying to catch up with the guns after the transfer, you can theoretically figure out who is taking the guns where and shut down back-alley channels before they can cause serious harm.
To wit, the government allowed otherwise obviously illegal gun sales to happen with the sole purpose of tracking the guns, rather than just shutting it down at the source. You can't find the leak if you don't turn on the tap.
In reality, this system was badly organized, and the folks running the guns were well organized, so guns that never should have been sold in the first place simply disappeared over the border. Now, bad system aside, F&F was plagued by two very serious problems
1) those who realized this was going very badly indeed were threatened with a battery of federal crimes if they called the Times and said "hey, the guys who are supposed to stop crimes are letting them happen and are seriously failing at everything they need to succeed at.
2) These guns vanished only to be found later, usually in the hands of dead or arrested cartel gunman, and were repeatedly traced to a long string of violent murders. This all came to a head when a Border Patrol agent was murdered with an AK-47 known to have been sold under the program, in the hands of a man known to the ATF as a suspect. A year later, same thing with an ICE agent, a gun known as part of the program used by someone who was already a suspect.
Basically, the ATF didn't move on the guns because they were still hoping to catch the big fish, and a lot of people got killed because of it.