Modulation instability describes the exponential amplification of a weak perturbation on top of a plane wave background and it has been extensively studied in the context of weakly nonlinear evolution equations such as the nonlinear Schrödinger equation (NLSE). It has been further suggested as a possible generation mechanism of rogue waves in the oceans. Here, we report on an experimental laboratory study in which the modulation instability dynamics have been seeded by random noise, added to a Stokes wave. Similarly to experiments recently performed in Optics, we show that the development of such spontaneous modulation instability in a water wave tank leads to the emergence of highly localized structures that can be interpreted as analytical NLSE breathers. Numerical simulations show very good qualitative agreement with the laboratory experiments.