The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of Turing-type instability for a nonlocal Fisher-KPP model. In particular, we prove that the solution of the considered nonlocal Fisher-KPP equation in the neighborhood of a constant stationary solution is destabilized via a diffusion-driven blow-up. It is also shown that the observed diffusion-driven blow-up is complete, while its blow-up rate is completely classified. Finally, the detected diffusion-driven instability results in the formation of unstable blow-up patterns, which are also identified through the determination of the blow-up profile of the solution.