Let's take a look at this graph just so we have a reference.
This shows the IR absorption bands of various gases found in the atmosphere. For this answer, let's focus in on the absorption bands of $\ce{H2O}, {CO_2}$, and $\ce{CH_4}$.
First, the answer to why methane is considered a worse greenhouse gas than carbon dioxide is largely because methane absorbs in regions where carbon dioxide does not absorb at all. That is, methane absorbs different wavelengths of light than carbon dioxide.
Another thing to note is that methane's absorption peaks overlap with some of the absorption peaks of water vapor. This is an interesting thing to note because the strength of methane's absorptions in those overlapping regions are much stronger than the absorptions of water.
So, we have two things which contribute to why methane is sometimes called a worse greenhouse gas than carbon dioxide. First, it absorbs in areas where $\ce{CO2}$ does not absorb. Second, these absorptions are very strong.
Something you might be interested in, however, is the fact that there are some people who make the argument that methane essentially doesn't matter as a greenhouse gas. This would be due to the overlap in absorption between water vapor and methane. For instance, methane has an average concentration of 1.7 ppm (parts per million) in earth's atmosphere. Compare this with the fact that the earth's atmosphere can be composed of 1% water vapor at sea level and .4% across the whole atmosphere on average.
Thus, the amount of water vapor in the atmosphere is much more than that of methane at any given time. So, it can be argued that we do not need care much about methane as a greenhouse has because water vapor will do all the IR absorption that methane would have done.
I don't know enough about atmospheric chemistry to really say if I agree or disagree with those claims. One thing I do know, however, which is interesting is the fact that the idea of global warming would seem to present more of a problem with water vapor than methane or carbon dioxide because as temperature increases, the ability of the atmosphere to hold water vapor increases which would accelerate the warming from water vapor, which would increase the amount of water vapor the atmosphere can hold...
Also, I believe that the metric which says methane is more dangerous than carbon dioxide is a comparison which does not account for other things which might absorb that light. In other words, the figure you quote do not account for the fact that water vapor might make methane irrelevant as a greenhouse gas.
Interesting Read: http://www.nasa.gov/topics/earth/features/vapor_warming.html
