While it's a noble idea but giving out free certs may give hackers easy way to exploit it. I'd be happy to pay $5 for a basic SSL cert knowing that it went through some kind of verification process via credit card.
Let's Encrypt works through installing a tool onto your web-server (or having your hosting provider do it for you); the main reason for this is automatic renewal of certificates (as they only issue 90-day ones).
However, it also allows them to verify your ownership of the domain; basically the program can do several things to your site that Let's Encrypt can then check, such as creating a file with a random name that they can look for at domain.tld/random_file.html.
This should be sufficient verification for free certificates, as an attacker would need to point your DNS records to their own malicious server, or gain access to your server and tamper with it, but these are things that a certificate wouldn't necessarily protect against anyway.
The point of extended validation certificates is to verify that there is someone legally responsible for the domain, so if money is lost or whatever you have someone to pursue legal action against, or report to trading standards etc.
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