There are several reasons why a Digital Single Lens Reflex (DSLR) camera has the POTENTIAL to make better pictures than a smaller sensor point and shoot. Note the emphasis. Buying a DSLR and getting better pictures is not a given but it does increase the odds and the abilities to get the shots you really want.
The one feature that made the single biggest difference in my photography when I stepped up to a DSLR was not the bigger sensor, the better lens selection, more manual controls or even the ego boost I got from finally shooting with a "real" camera. Yes, that is actually one of the benefits of owning a DSLR. Our belief has a way of shaping our reality. The early belief that my DSLR automatically made me a better photographer was instrumental in helping me become a better photographer. It also helped me realize very quickly that my former camera had not been my biggest limitation. :-)
So what was the feature that made the biggest difference in my photography? The hot shoe - That little bracket on top of the view finder where you attach an external flash, or at least some kind of trigger to light one. You can buy small sensor cameras with a hot shoe on them but they are going to cost at least as much as some of the entry level DSLR's so unless the smaller size is important to you, you may as well invest in a system that will give you maximum flexibility. As with all gear in this addictive hobby/profession, you can spend a little or a lot on lights. I've invested plenty myself but I think I only paid $15 for the off camera flash I use most often, a Vivitar 4600 with a vari-power slave module on it. More on that light later...
Take a look at these two lighting lessons on how to use the hot shoe to your biggest advantage. Like so many up and coming photographers, I owe a huge debt of gratitude to David Hobby for helping me understand the power and use of off camera flash. Fair warning, the way of the Strobist is addictive. Once you learn to use an off camera flash you will be hooked on well lit photography for good. Here is just about everything you need to know on the subject from a true master of the craft:
Strobist Lighting 101
Strobist Lighting 102