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 .
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The ripening itself is caused by ethylene gas released by the food, which is trapped by any kind of bag.
As far as I know, there's nothing inherently special about a brown paper bag, other than the fact that it's porous and thus still allows some air to get in and out.
Plastic ziplock bags have no ventilation, so they don't work nearly as well.
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The secret is that the fruit produces ethylene (a hormone found in plants), and ethylene promotes ripening.
By placing fruit in a paper bag the ethylene collects (rather than dispersing in the room), increasing the concentration around the fruit and speeding ripening.
About.com has an interesting article on ethylene and fruit ripening called Fruit Ripening and Ethylene Experiment .
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More importantly, brown paper bags allow moisture to pass through, so you don't get condensation that will accerate mold growth.
Just leave produce on your counter in a plastic bag and witness the flexible petri dish!
White would probably work if it is porous enough.
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Some types of fruit ripen with increased ethylene production and a rise in cellular respiration (the ripening fruit draws in oxygen and gives off ethylene).
This happens in "climacteric fruit": Apples, bananas, melons, apricots, and tomatoes, among others (citrus, grapes, and strawberries
are non-climacteric--you can do a search for "climacteric fruit" to see which are and which aren't).
The ethylene produced during respiration enhances the ripening process so when we put climacteric fruit in a paper (or plastic) bag, "none" of the ethylene can escape, thereby exposing the fruit to more and more ethylene as it continues to ripen.
Paper bags only restrict oxygen flowing in/out of the bag while plastic prevents essentially all oxygen flow.
Since climacteric fruit need oxygen for respiration, closed plastic bags will limit the amount of respiration to that allowed by the oxygen trapped in the closed plastic bag.
Opening a plastic bag will allow some oxygen in (while letting some ethylene out)...in that case fruit will ripen faster with an open plastic bag than fruit just sitting on the counter but will ripen slower than in a paper bag because the paper keeps more ethylene in while still allowing oxygen in.
The short answer is that paper bags keep the ethylene trapped in the bag which enhances the ripening process while at the same time permitting some oxygen to enter the bag, allowing the fruit cells to respirate and produce more ethylene.
Assuming brown paper bags are the same as other paper bags, then this should be true for brown paper bags as well.
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I often see this technique suggested as a way to speed ripening. Why does it?  Also, is there any data available on how fast it ripens comparatively?
Why does a brown paper bag speed ripening?
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<storage-method><food-science><ripe>
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Brown paper bags help to trap and concentrate the levels of ethylene gas in fruit that produces it, while allowing moisture to pass through without causing mould growth.  Placing fruit in a bag concentrates the ethylene gas which causes fruit to ripen. The paper bag allows air to pass and stops condensation.
This will only work for certain types of fruit. Placing fruit in a bag helps to concentrate the ethylene gas which causes them to ripen. Placing the fruit in a paper bag allows air to get in and prevent condensation.