What's being exploited?
The attacker is exploiting a combination of inadequate pricing (too cheap) for specific EVM operations and inefficient client implementation of that operations (mainly geth which is the most widely used one).
This specially crafted "bad transactions" cause a lot of I/O on the clients, resulting in many of them not being able to keep in sync with the blockchain. This also resulted in block times going considerably higher then the average 15s at times.
More details here.
Can it be stopped?
I think so. The protocol changes in the fork will make this kind of "transaction spamming" much more expensive. Even if the attacker decided to pay the much higher price and continue, the effect should get smaller, because additional measures are being implemented, e.g. automatic lowering of gas limit after "slow blocks" (with the result of expensive transactions not being processed) and optimizations to the I/O related code.
More details about protocol changes here.
Of course all this doesn't guarantee the spamming to stop. If the attacker has deep enough pockets and is determined enough to harm the Ethereum project, he/she may continue to annoy the Ethereum community.