How does best response imply indifference? I recently read Leyton/Brown's Essentials of Game Theory and I keep coming back to this (Sec. 2.2):

When the support of a best response $s∗$ includes two or more actions,
  the agent must be indifferent among them—otherwise, the agent would
  prefer to reduce the probability of playing at least one of the
  actions to zero. But thus any mixture of these actions must also be
  a best response, not only the particular mixture in $s∗$. Similarly,
  if there are two pure strategies that are individually best responses,
  any mixture of the two is necessarily also a best response.

This is followed by an example on the next page of using this observation as the basis of computing a mixed-strategy Nash equilibrium.
I get it, but I don't get it "in my gut", which is troubling. It just seems to sound highly counter-intuitive that having a "best response" implies that we are thus "indifferent" in our response. I've tried to make this feel natural, but I haven't succeeded.
Does anyone have a better explanation/analogy/picture by which this can be made memorable and intuitive?

Here is an intuitive answer. Imagine a box of chocolate pralines: some are dark chocolate, some are white chocolate, some have a nougat filling, and so on.
Your "best response" is picking the flavour you like the most; say, the nougat-filled ones. 
Because you are indifferent among any of the nougat-filled pralines, your best response is to pick any of them. If randomised strategies are allowed, any probability distribution over the nougat-filled pralines is also a best response. Choosing randomly over your preferred pralines works as good as choosing one of them. (This may even be what happens in your brain when your hand picks one of the nougat-filled pralines.)
