Intuitively, why does a lack of clustering standard errors lead to erroneously smaller standard errors? Looking at the calculations and seeing the difference between SEs that are clustered and not clustered gives me an idea of the mathematics, but not really an intuition.
Suppose I sample 3 people from the population of CV users, and I just happen to include Student, Kjetil, and myself. Now I take another sample, which just includes myself three times (perhaps sampled on different days). I think you would agree that in some sense the second sample contains fewer observations, even though $N=3$ in both cases. My usage behavior over time may be very correlated.
SEs are proportional to the sample size. Clustering is a way to formally adjust for the fact that observations in the same cluster may have correlated errors (though correlation in errors across clusters is still precluded), and that sample is "smaller" and contains less information.
Note that it is also possible for errors to shrink when the within-cluster correlation is negative. This is fairly rare, at least in social science applications.
Obviously sampling myself three times is silly, but if you are sampling clusters of farmers that all live in the same village, you are moving closer to the first type of sample, though some less strong correlations may remain. This correlation may describe some unobserved shared local event, like a swarm of locusts descending on some villages but not others. It is in the error since locusts are unobservable for the statistician. So local shocks like locusts induce agricultural output of a pair of farmers from the same village to be correlated. If locusts don't travel too far, output between a pair of farmers from different villages will be uncorrelated. Or there might be some region specific things, like government education or agricultural subsidy policies, so in that case you would want to cluster at region rather than village level. This assumes that all the villages in the same region have the same policies.