Eigenvectors are "important" directions with respect to some linear transform (operator). The eigenvalues are constant scaling factors that relate input vectors (inputs to the operator) to output vectors.

Knowing that an operator merely scales (but doesn't rotate) certain input vectors is an important property to characterize it.

Imagine some matrix describing the stresses (strains?) in a continuous medium of material, about a certain point. If you can find special directions from that point where the forces are just scaling, then you know you're just dealing with tension/compression in those directions, not bending.

Same thing applies for fluid flows, it can be useful to find out where the flow is just speeding up / slowing down, but not twisting.

... helpful?