The implicit "definition" of a beard as an arbitrary number of facial hairs is the primary issue. The proof starts with a very easy to grasp fact; 1 facial hair does not a beard make. A variable $k$ is then given the value 1, and the number of facial hairs an individual has, $n=k=1$, is not a beard.
By adding one hair to $k$, we don't change the answer; two hairs is still no beard. Three hairs, same thing. The proof then concludes that, for a given number of hairs n, because $n=k=1$ is no beard and $n=k+1$ doesn't change the answer, there is no $k$, produced by incrementing from the base case, that constitutes a beard. The "proof" is essentially stating that an arbitrary and undefined number of hairs $x$ is needed to make a beard, and $n=k<x \therefore n\neq x$ for all $n$.
It's a "boiling a frog" argument; drop a frog in a pan of boiling water and he'll hop out. But, put him in a pan of cold water, and heat it up by degree, the frog will just sit there, because each degree feels the same as the last. There is, however, a qualitative difference between water that is 211*F, and water that is 212*F, that does not exist between any two adjacent degrees of water temperature < 212 (until you get to 32). By the same token, somewhere between one hair and 8 million hairs, you'd call it a beard; perhaps it's scraggly, perhaps it's just a goatee, but a goatee is a beard and a scraggly beard is a beard.