One way to do that is to log all exchange or transmission transactions above a certain size -- which is what licensed money changers and transmitters have to do. You're not prohibited from performing these transactions, you just have to give them your name and SSN or Taxpayer ID.
Sure, there are ways around this, like buying a million rubber duckies in China, then shipping them to the US and selling them at carnivals. But -again- try to import a million of anything, and US Customs will want to know your name and SSN.
If you're talking about changing $100 or even $1,000 into BTC, the government doesn't really care. You might think, can't the criminals just split their transactions into small enough chunks that they can avoid the law altogether? Sure, maybe they could (and certainly they do) -- but at some point, having to conduct a thousand individual exchange transactions to move a million bucks starts to get onerous, and is also likely to leave a data trail -- there will be correlations somewhere.
It's never going to be impossible to exchange BTC for USD untraceably. The point here, which should come as no surprise, is that the government intends to make it impossible to exchange large amounts of value anonymously without it being annoying, time-consuming and costly.