I've heard great things about Let's Encrypt, many people who I trust use and trust Let's Encrypt, and I use and trust them too.

But beyond that warm fuzzy feeling, and what they write about themselves on their web site, I'd love to know more particulars about the nature of why they're different, why so many people trust them, how they came about, who's running it, who's supporting it, who are they competing against, and what their mission and back story is. Why are they pushing other CAs to up their game, and why weren't those other CAs doing what Let's Encrypt is now pushing them to do, in the first place?