Why does a brown paper bag speed ripening?
Placing fruit in a bag will help hasten the ripening for only a few fruits.  Most fruits will not ripen (ever) once they have been picked.  The only fruits that ripen once they're picked are bananas, avocados, pears, mango, and kiwifruit.  Ripeness in fruits is based on sugar content, not color, thus most tomatoes in grocery stores have no flavor because they were picked green and exposed to ethylene gas to induce color change.  Tomatoes are what are known as a "climacteric fruit" which means they'll change appearance based on climactic conditions but they don't ripen.  Yes, they continue to soften but that's simply the process of cell deterioration know as decomposition.

Some fruits such as peaches, plums, etc. will seem to develop more sweetness as they sit on the counter but that's because they're also losing moisture and the residual sugars are concentrating in the cell walls.

Placing fruit in a paper bag helps to concentrate the levels of ethylene gas which is what helps induce the ripening of the above mentioned fruits (bananas/avocados, etc.). In fact, avocados and pears must be picked in order to ripen.  Pears that are left on the tree will simply rot.   As mentioned previously...  brown paper bags used to be something everyone had around their house so it was a commmon item before the switch to plastic bags. 

Have you ever noticed some people like red bell peppers and not green ones?  The reason is due to ripeness.