See the wikipedia article on the German tank problem for a good story leading up to why you care about them, or for an even simpler one:
Suppose there's a bucket full of N tickets, labelled 1 through N. You don't know N, but you pull out a ticket at random, and see that it has n written on it. What do you suppose N is?
The maximum likelihood estimate says n, which intuitively, to me, is utter stupidity. The Bayesian story sort of confuses me, too. Suppose I had a prior belief that N is uniformly randomly chosen between 1 and 100, and suppose I draw n=2. Then I can rule out N=1, and I believe then that
Pr(N=3|n=2) = Pr(n=2|N=3)Pr(N=3)/Pr(n=3) = (1/3)(1/100)(1/(1/100 + 1/200 + 1/300)) = 1/(3 + 3/2 + 1) = 2/11
[This calculation is utterly wrong, thanks to Susan for pointing it out, still the conclusion I drew from it by coincidence is still right, which is:]
In that case the most probable size of N is allegedly still 2, which seems wrong.
But the MVUE says we should estimate N = 2n-1. Clearly being inutitively right is not the same thing as being formally right, but I'm compelled to consider MVUE an interesting idea at least because it delivers intuitively correct answers when other formal things I already know about fail to.