How to explain intuitively to a lay audience that the variance is an unbiased estimator? I have data for the concentration of several chemicals in the milk of 10000 cows and have to explain to policymakers and the lay public (i.e. people with no or limited knowledge of statistics) that the variance calculated from our sample is an unbiased indicator of the variance in chemicals' concentration in the population.
Call it professional deformation but I am having a hard time coming with an accessible explanation. What is an intuitive way to explain that the variance is an unbiased estimator?

It is rather odd that you would need to explain the concept of unbiasedness to a lay audience at all.  If they are not already familiar with the idea of an "expected value", and some other general ideas in sampling theory, then what exactly is the necessity in distinguishing a biased from an unbiased estimator?  Will you also explain consistency of the estimator, etc.?
A far more important explanatory and interpretive problem I see here is specifying what you mean by "the population" and how you sampled from this population.  What set of cows (larger than your sample of 10,000 cows) is "the population" here?  Did you use simple random sampling from this population, or some other sampling method?  Are you absolutely sure that your sampling method does not induce bias?
In any case, if you really do need to give an explanation of the concept of an "unbiased" estimator to a lay audience, something like this would not butcher the concept too much:

We have estimated the variance in the population for our analysis, which was [specified set of cows].  We estimated the variance of the concentration of chemicals in these cows using something that statisticians call an "unbiased estimator".  Roughly speaking, this means that our estimate of the variance will ---on average--- equal to the true variance in the population.

