The benefit is that, whereas with double-entry accounting, when one of the receipts is tampered with in a professional way, you cannot tell anymore which receipt is the correct one. But in triple-entry accounting, the public receipt (the blockchain) serves as a tiebreaker and provides the truth that all can (or need to) agree on.
It's just a modification on standard accounting that has nothing to do with cryptography. I could be wrong, but it really seems like the author heard someone use the term "triple entry accounting", googled it, and assumed this guy Yuji Ijiri must have been way ahead of his time without finding the book to actually confirm it.
I understand the main point of this article is just how great the blockchain is, but it shouldn't be framed around a concept that the author apparently did no research on and isn't able to explain. Huge portions of this article are just flat-out wrong, for example:
"The Dawn of Triple-Entry
Most people missed Professor Ijiri’s breakthrough because it straddles two equally obscure and poorly understood fields: cryptography and accounting. It’s rare enough to find a person who’s versed in one of those disciplines, never mind both. Without that kind of interdisciplinary understanding, it’s no surprise that his invention went over like a lead zeppelin. There’s also the little problem that he was incredibly ahead of his time. Encryption had not yet entered the public consciousness. If you work in information technology you might remember the Clipper chip scandal, where the NSA tried to mandate a backdoor in all encryption. That was in 1993. Ijiri published his work in 1989. It passed mostly unacknowledged by the general public. Then, in 2006/2007, a self-taught programmer likely stumbled on the system. He was working on an alternative currency, with no centralized trust. It was called Bitcoin."
Literally none of this is true, even the summary of the Clipper chip scandal. A quick skim of the Wikipedia article on the blockchain tells you that Satoshi was not inspired by an obscure accounting paper, but by previous work on blockchains and cryptocurrencies.
The idea is that, as with single to double, triple entry adds another record.
In this case it's the blockchain but it actually doesn't have to be a blockchain, that's just a method for having a publicly verifiable (and cryptographically reliable) 3rd entry.
Long story short: it's not really about bitcoin/blockchain but more how those technologies enable a really interesting and potentially game changing accounting process.