So, the author is trying to say, with blockchain, we can collectively trust what happens online. But what on earth does "to collectively trust what happens online" mean?
There are multiple meanings that you could attach to that, but here's the one I think is the most true/useful/obvious.
To track what happens online, we use computer programs and databases. What happens when those programs disagree about what the state of the system is or ought to be?
For example, an online auction receives a bid moments before it closes. The bidder who sent the bid claims it was timely. The bidder who would lose the auction if the highest bid was accepted claims it was a second too late. The seller doesn't know, because ze was offline when the auction closed.
The typical solution to this is to pick some trusted computer, and let it decide. In this example, that would be eBay's database server. (Or, in some systems, a quorum of eBay's database servers.)
Blockchains are a search for a better solution.
Does it mean everyone can trust whatever is recorded on the blockchain as the underlying technology makes whatever that is recorded on the blockchain trustworthy?
For example, if you recorded a claim to be King of the United Kingdom on the blockchain, that would not mean that you were the King.