In the figure you can clearly see that the compressive loading (indicated by the blue arrow) has caused deformation in the cylinder so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change, thus the material is forced out laterally.
See also: Modulus of elasticity, Elastic, Inelastic, Visco-elastic (a.k.a. Plastic).