Bitcoin took off for pretty much the same reason as why bittorrent took off.
Yesterday someone made angry comment about my characterization of Bittorrent. He said that Bittorrent has other advantages than just the ability to share copyrighted files anonymously. Of course that is true. The server distributing a bittorrent file can shift the burden of sharing the file to the first users who download a copy from his server. So, yes, you can save bandwidth on your server by using Bittorrent. However, that is not the reason why the use of Bittorrent took off.
The reason why bittorrent took off so spectacularly, is because it facilitates anonymous sharing of copyrighted files.
Bitcoin rapidly took off for a similar reason.
Imagine that person A wants to supply person B with goods who will pay money to person A for such delivery. Both persons are located at a large distance from each other. If B pays A using the banking system, then there will be traces of the transaction and their identities will be recorded. How do they solve the problem?
-  Person B exchanges fiat money for bitcoins on his local exchange market with person C1.
-  Person B signs over the bitcoins to person A.
-  Person A exchanges the bitcoins back to fiat money on his local exchange market with person C2.
-  Person A sends off the goods to person B
In other words, the use of bitcoin replaces one fiat money transfer by two bitcoin/fiat exchange operations. This is attractive. While the fiat money transfer would link A to B, the exchange operations link A to C2 and B to C1:
(A,B) –> (A,C2) + (B,C1)
For quite a few trading parties (A,B) this is a strongly-desired anonymization step.
For the system to work, it is only necessary that bitcoin manages to keep approximately the same exchange value between exchange operation (B,C1) and (A,C2). Since bitcoin usually manages to keep the same exchange value for a few hours in a row, it is certainly suitable for this type of transaction.
Concerning the exchange markets, it does not matter that they could keep a record of the fact that A has traded with C2. It is useless information, since it does not disclose the real, underlying link between A and B.
Of course, there is a trace in the blockchain that shows:
C1 –> B –> A –> C2
This is less of a problem than it looks like. For a starters, it is hard to track and trace single-use, throwaway bitcoin addresses. Secondly, it is not particularly hard to break the chain:
C1 –> B –> X1 X2 –> A –> C2
For example, B exchanges his bitcoins with X1 in exchange for litecoins. B exchanges his litecoins with X2 in exchange for bitcoins again. Now the chain is solidly broken. There is simply no way left to follow the chain from C1 to C2.
While a bitcoin to fiat exchange market will keep track of the identities of their customers, crypto-to-crypto markets tend to be entirely anonymous. If they are not, few customers will use them, because crypto-to-crypto markets are the most natural way to launder bitcoins.
Of course, bitcoin is useful for other purposes than anonymous fiat money transfers, but these other reasons would never have caused Bitcoin to take off the way it did. At the core of Bitcoin’s success, there is a strong desire for privacy. No other mechanism offers the same features. Bitcoin is the clear winner in this realm.