When you run
add-apt-repository to add a PPA, this gives you a cryptographic guarantee that the content of the packages that you install are the ones from the PPA that Launchpad (the Ubuntu service that distributes PPAs) considers to be this PPA — in this case,
add-apt-repository ppa:webupd8team/sublime-text-3 guarantees that, as long as Ubuntu's servers are secure, the packages that you download from that particular source were really from the Launchpad user
webupd8team. So you are protected from a MITM attack.
add-apt-repository downloads a public key associated with the PPA over HTTPS. The authors of the program don't need to provide a checksum because APT will check the signatures. The only piece of information that you need to retrieve securely is the name of the PPA (especially the name of the PPA user). A phishing attack could try to convince you to install from
ppa:webupd8teteam/sublime-text-3, for example, instead of the legitimate one.
Launchpad does the building, too, so if you trust Ubuntu's servers (and the authors and providers of the compilers and other build tools), you can also trust that the binary implements the source code uploaded by the PPA author.
This does not protect from an attack on Ubuntu's build servers: if they're breached, the attacker could inject a backdoor at build time. This doesn't protect from an attack on Sublime's own infrastructure that allowed the attacker to inject their source code, nor of course from a malicious developer who works for Sublime and could have included a backdoor in the genuine source code.