Is there a good explanation on how bitcoin addresses the problem out there? I haven't come across a simple explanation that validates the approach (in my mind at least).
This doesn't answer your question specifically, but:
For the Bitcoin network to properly operate, it requires at least 51% of the computing power of the network to be "good", well-behaving nodes. That 51% means 51% in terms of block-mining. So, computing power in this case means "ability to find plaintext that results in a certain double-SHA-256 digest".
Therefore, Bitcoin has not come across a completely rock-solid solution (as with many cryptographic protocols), because a determined attacker or group of attackers could theoretically achieve control of the network with sufficient computing power. At this stage though, it's difficult enough to basically be considered infeasible, unless a global superpower tried to tackle it.
