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It lets you say something about the p-part of the ideal class group. The p-adic class number formula relates the values of p-adic L-functions with the class number, along with the p-adic regulator. This relation is what allows us to say that p divides the class number of the cyclotomic field Q(zetap) if and only if it divides the numerator of a certain Bernoulli number.
If you want to know more about this stuff, then you should pick up Washington's Introduction to Cyclotomic Fields.