Do you want the cynical answer or the idealist answer?
The cynical answer is that banking institutions and oligarchs are adopting cryptocurrencies because with the various recent leaks and the (long overdue) political willpower in the West to do something about tax evasion, they need something more reliable than physical tax havens.
One of the reasons why I suspect the current BTC rate is not just a bubble is that it matched the recent crackdown on corruption in Saudi Arabia. Hundreds of billions of dollars are at risk of being lost. We have seen similar hikes after Cyprus banking fiasco and, IIRC, after a round of Russian sanctions. Still, when even my step-mom considers investing in BTC, there is clearly a speculation bubble going on over this, but I doubt the crash would bring it to pre-december levels.
The idealist answer is that blockchains allow to secure transactions that would normally require a trustable third party. If you want to run a virtual currency you normally need a database somewhere that says "Bob owns 176 dogecoin, Amanda owns 9871 dogecoins". The owner of that database can theoretically suddenly remove or add amounts on accounts in the database without anyone else's authorization. On the blockchain, the database is shared by thousands (millions now I guess?) of nodes constantly checking each other. To make a fraudulent modification and get away with it you need to own 2/3 of the network.
Currencies were the first things implemented but now some smart contracts have been added: contracts that are programmatically defined and that the blockchain will ensure execute if some specific conditions are met. E.g: "If the account #25676 contains more than 10 BTC at the first day of a month, split its amount between these two different accounts". This gave rise to the possibility to create enforceable shareholder contracts without having to trust a legal system.
The idealists who put that in place are called cryptoanarchists. (Warning: Bias ahead) simply put, these are people who think that big institution will always fail. That corruptable systems will be corrupted and that the people creating crucial infrastructure today are so flawed that the general demise is inevitable. They see a big value in the fact that we don't need to trust an entity to do transaction. I used to think that way too.
However, fact is that entities whose business is to be trustworthy tend to do a good job 99.99% of the time. The 0.01% of the time is when you are called Wikileaks or Khodorkovsky. If you are, then you put a lot of value in the lack of third parties, but if you are part of the 99.99% you probably don't.
I still like the fact that blockchains are creating competition for other institutions. The speed and low price of transactions (well, now it is getting higher in BTC, hopefully this will be solved) puts the big institutions to shame. Recently a 10 minutes SEPA transfer initiative was started, probably because of this. It is the main transfer system in Europe and it would typically take several days to process.
The fact that companies could exist solely on blockchains and be created in a matter of minutes will hopefully put a similar pressure on administrations.