If your site is good on a mobile device, it translates better to all devices. More important, though, is that the mobile-first approach is also a content-first approach. Mobile has the most limitations, screen size and bandwidth to name a few, and so designing within these parameters force you to prioritize content ruthlessly.
The mobile-first approach organically leads to a design that’s more content-focused, and therefore user-focused. The heart of the site is content — that’s what the users are there for.
One caveat, though, is that mobile users sometimes require different content than desktop users. Device-specific content can be gauged by considering context — what, in a given situation and a given environment, will your user appreciate more. The best way to plan ahead for these is creating user scenarios.