Stick-slips are spontaneous, unstable slip events during which a natural or man-made system transitions from a strong, sticking stage to a weaker, slipping stage. Stick-slips were proposed by Brace and Byerlee (1966) as the experimental analogue of natural earthquakes. We analyze here the mechanics of stick-slips along brittle faults by conducting laboratory experiments and by modeling the instability mechanics. We performed tens of shear tests along experimental faults made of granite and gabbro that were subjected to normal stresses up to 14.3 MPa and loading velocities of 0.26-617 micron/s. We observed hundreds of spontaneous stick-slips that displayed shear stress drops up to 0.66 MPa and slip-velocities up to 14.1 mm/s. The pre-shear and post shear fault surface topography were mapped with atomic force microscopy at pixel sizes as low as 0.003 micron^2. We attribute the sticking phase to the locking of touching asperities and the slipping phase to the brittle failure of these asperities, and found that the fault asperities are as strong as the inherent strength of the host rock. Based on the experimental observations and analysis, we derived a mechanical model that predicts the relationships between the measured stick-slip properties (stress-drop, duration, and slip-distance) and asperity strength.