The key thing in our blockchain is the storage contract, which is something like a blockchain-SLA. The host has an obligation to store some data, and the host also puts up money out of pocket as a promise that the data will be stored faithfully. The blockchain will occasionally challenge the host, and if the host cannot create a proof that the data is still being stored, the host's collateral is forfeit.
The renter also puts money into the file contract, this money is used to pay the host if the host is honest/keeps the data, and the money is destroyed if the host is dishonest or loses the data. The money gets destroyed so that the renter has no reason to interfere with the host, nothing to gain from seeing the host fail or working to make the host fail.
Thanks to the blockchain, the host has a guaranteed payment if they store the data, regardless of whether the renter sticks around for the duration of the contract or not. And also thanks to the blockchain, the renter knows that the host will not get paid until the end of the contract, and that they will actually lose some of their own money if they lose the data.
There's another great benefit though to cryptocurrency, which is that payments are super low friction. Every time you upload data to the Sia network or download data from the Sia network, you make hundreds to thousands of tiny payments to the various parties over state channels. This wouldn't be feasible using more traditional payment systems, we'd have to track everything in a centralized database, and that actually doesn't scale very well.
Sia is both low cost and high speed, but these are side benefits over the primary goal of uncompromising decentralization. Storage just happens to be an application where the decentralized version is actually highly competitive to the centralized versions.
I'm glossing over a fair amount here, if there's anything that doesn't make sense or seems off feel free to ask, I'm happy to clarify.