Aside from looking great, the benefits of responsive web design are vast. Namely this; mobile will soon overtake desktop for Internet use.
The link here is that with responsive website design, you create a great-looking website that responds (hint hint) to individual Internet platforms in the same way.
Let's put it this way, you know when you visit your favorite website on your tablet or smartphone, it looks different, you have to zoom and not all the pages are available? Well the theory behind responsive website design means all of that inconvenience goes out the window.
Your website looks the same on desktop, tablet, smartphone -- all devices. One URL, one HTML code, one CSS query and one set of content - as demonstrated on this infographic.