Some intuition is afforded by the Jordan Normal Form, and even more is obtained by understanding a proof.
The Jordan form of an endomorphism decomposes it as the direct sum of "Jordan blocks"; if you have an intuition for what a block is doing, you can understand the entire direct sum.
A Jordan block geometrically is the sum of two operations. One rescales everything by a constant, the eigenvalue. The other essentially collapses the module onto a codimension one submodule.
The collapsing is the nilpotent part. A geometric intuition is obtained by considering the simplest nontrivial example, the linear map defined (in coordinates) by $(x,y) \to (y,0)$: two dimensions are collapsed onto one essentially by forgetting the first dimension.
In a Jordan block, the nilpotent operation is generally $(x,y, \ldots, z) \to (y, \ldots, z, 0)$. This establishes a hierarchy in the module: the last dimension generates the kernel of $T$, the last two dimensions are killed by $T^2$, and so on. Iterating $T$ enough times eventually produces the zero map--that's what it means to be nilpotent. Thus the kernel of $(T-\lambda I)^k$ picks up all the Jordan blocks associated with eigenvalue $\lambda$ and, speaking somewhat loosely, each generalized eigenvector gets rescaled by $\lambda$, up to some "error" term generated by certain of the other generalized eigenvectors.
In two dimensions, then, a Jordan block effects a transformation that can be written in suitable coordinates (given by two of the generalized eigenvectors) as $(x,y) \to (\lambda x + y, \lambda y)$: an isothety and skew transformation together. The geometric picture isn't really any different for higher-dimensional blocks.