The local structure of La(Fe1-xMnx)AsO has been investigated using temperature dependent Fe K-edge extended x-ray absorption fine structure (EXAFS) measurements. The EXAFS data reveal distinct behavior of Fe-As and Fe-Fe atomic displacements with a clear boundary between x <= 0.02 and x > 0.02. The Fe-As bondlength shows a gradual thermal expansion while the Fe-Fe bond manifests a temperature dependent anomaly at similar to 180 K for x > 0.02. It is interesting to find characteristically different nature of Fe-As and Fe-Fe bondlengths shown by the temperature dependent mean square relative displacements. Indeed, the Fe-As bond, stiffer than that of the Fe-Fe, gets softer for x <= 0.02 and hardly shows any change for x > 0.02. On the other hand, Fe-Fe bond tends to be stiffer for x <= 0.02 followed by a substantial softening for x > 0.02. Such a distinction has been seen also in the As K-edge x-ray absorption near edge structure, probing local geometry around As atom together with the valence electronic structure. The results suggest that local atomic displacements by Mn substitution inducing increased iron local magnetic moment that should be the main reason for its dramatic effect in iron-based superconductors.