How do I divide the columns of a matrix by the sum of its elements?

You can use Normalize with its second argument for this purpose:

(mat = Normalize[#, Total] & /@ Transpose@L // Transpose) // MatrixForm

Instead, if you were normalizing the rows by the sum of their elements, you could simply leave out the transposes and do

mat = Normalize[#, Total] & /@ L

or even

mat = #/Tr@#& /@ L

For your specific problem (transition matrix), you can use the new Markov process related functions in version 9 to get the transition matrix:

With[{m = DiscreteMarkovProcess[, L]},
   mat = MarkovProcessProperties[m, "TransitionMatrix"]
] // MatrixForm

If you need to do this with all columns, then:

Transpose[#/Total[#] & /@ Transpose[L]]

Why transpose when you don't have to?

#/Total[L] & /@ L

(Just resurrecting this for a bit of "code golf.")

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