Being non-commuatable operators, momentum and position cannot have simultaneous eigenfunctions. But in "Theoretical Minimum: QM" by Lenny Susskind and Artsy Friedman, in explaining Heisenberg's Uncertainty principle: 
I don't understand why they assume that the eigenstate of momentum (Psi with subscript p) is going to have a localized poistion component. I would love some clarification on this topic. Thanks!
You confuse different things. An eigenstate of p or x is defined with the equation $A \psi = \lambda \psi$, where A is the momentum/position operator and $\lambda$ the corresponding eigenvalue. Psi here is an abstract mathematical element of the QM Hilbert space. What you have written down here as $\psi_p(x)$ is an eigenstate of the momentum operator in the x representation. It is a specific representation of the Hilbert space vector, but not an eigenstate of the x operator.
