Fenixgoon did a good summary. But, I'd like to elaborate a bit.
With anonymous transactions, someone has to be a trustworthy source. I can't just say that "hal2kilo gave dullard all his money" and take everything you own. You would respond with something along the lines of "dullard is lying". How would the world know who is correct? Who should get all of hal2kilo's money?
With cash, that trustworthy source is usually a bank or it could be a government. You tell a bank to send me all of your money. My bank accepts. All is good since we trust the banks to decide that you legitimately sent your money (large transactions go through detailed fraud checks before being sent).
But with crypto, people decided that they can't possibly have a bank be the trustworthy mechanism to decide if transactions are legitimate. There was no real strong universal reason for this, but there are fringe reasons that banks are not good for everyone (there are numerous unbanked people). So, some other source of trust had to be created. Proof of work is one possible method. Basically, if you are wealthy enough to buy lots of equipment and waste lots of resources, then you must be trustworthy. I can't quite agree with that logic that wealthy people are more trustworthy than banks, but proof of work does function in general. It just is impossible to keep up. There is not enough equipment nor enough energy in the world to do it.
Proof of stake is another possible method. If you have invested enough into the crypto, then supposedly you want to do good and are trustworthy (why would you want to destroy the money you invested in that crypto?). Again, though, it assumes the wealthy are the trustworthy ones because they are the ones with large shares. Also, it has a major flaw. If ~49% of the owners currently online say one thing, they win--whether they are telling the truth or not. Cut a few international cables and suddenly even a small owner can have 49% of the currently online shares and suddenly "hal2kilo gave dullard all his money" is the truth because I said so.
There are other methods, but those are the main two that are being tried. I personally would just go back to banks since it is cheap and works with a lot less "the wealthy are the trustworthy people" hogwash. (Consider for a moment if a bank or a government was wealthy, then we ultimately just put all our trust in that bank or government).