In the homogeneous (or transient) response of a given linear system, each pole (or complex pair of poles) contribute terms of the form exp(a*t) where "a" is the value of the real part of the pole in question. If these poles are in the left-half plane, i.e. the system is stable, then the contribution to the response for a given value of "a" dies out faster if the pole is farther to the left (since exp(-a*t) approaches zero faster for larger "a"). Therefore, the poles closest to the imaginary axis are the ones that tend to dominate the response since their contribution takes a longer time to die out.