In short, you (publisher, developer, and designer) should care because you want the visitors to your website to have the best experience possible, without forcing them to adapt themselves.
There are essentially two ways you can give your audience a good experience utilizing responsive design:
The first is optimizing the layout of the content.
If a user is browsing from a mobile phone, they generally don’t have a lot of screen real estate to work with. Phones today will typically zoom out automatically, so that the entire website can be seen onscreen. This can be good, as it gives the reader access to the entire site, but it can also be frustrating when trying to find information that is located in a tiny part of the upper right of the screen. If you could move some things around, make some things bigger and not have as many columns you’ll give your mobile reader a much better experience.
The second is to adapt the content that is shown.
If you own a restaurant and a potential customer is browsing your site from a mobile phone, chances are they aren’t that concerned with how pretty your site is — your foodie blog with the awesome slideshow of delectable dishes scrolling from side to side isn’t very useful in that situation. They want to know what your hours are, where you’re located, how to make reservations, and want a look at the menu.
Your potential customer browsing from a desktop computer probably isn’t looking to eat right now, and isn’t in a hurry to see where you’re located and what your phone number is. Most likely he’s looking to see if you offer a good atmosphere and what types food are available.
These are obviously generalizations but you can see the benefits of having differing content presented to people in different screen viewing circumstances.
Mobile responsive design takes care of this all “on the fly”, and without multiple versions of your site to maintain