To me the real value is that for a very fair price I no longer have to be concerned with the ins and outs of sending email. Once you start sending tens of thousands of emails you run into problems. People don't receive the emails. They get blocked. Users accidentally mark them as spam. And so on and so on.
I don't want to focus on figuring out email - I just want to send them and have them get to the users that want them.
I use MailGun for email for the same reason that I use FreshBooks for invoicing my clients — because I want to spend my time building my business rather than focusing on solving a problem that someone else has already done a great job of solving.
This is a great example. I think some of the disconnect with other commenters is thinking that sending email is trivial. It isn't. Even at small scale, deliverability starts to become a problem, there is infrastructure management (for example, this why people use heroku rather than just AWS) and receiving and tracking email requires managing all sorts of email weirdness that are not addressed by using software libraires (this is analogous to html parsing - getting all the non-standard or ambiguous cases right is hard).
I don't want to focus on figuring out email - I just want to send them and have them get to the users that want them.
I use MailGun for email for the same reason that I use FreshBooks for invoicing my clients — because I want to spend my time building my business rather than focusing on solving a problem that someone else has already done a great job of solving.