Skyrmions, swirling spin textures with topologically protected stability and low critical driven-current density, can be generated from the stripe domain with current pulses, bringing them closer to practical applications in racetrack memory. However, the mechanism of this topological transition from the stripe domain to the skyrmion remains unclear because the transition process occurs at a nanosecond timescale, giving rise to difficulties in observing this process using imaging tools. In this study, we controlled the domain wall - skyrmion transition by combining Joule heating with spin-orbit torque (SOT) and experimentally observed the details of this process, by which we confirmed the mechanism: the spatial variation of the topological charge density induces half skyrmions branching from the stripe domains, and these half skyrmions overcome the surface tension and break away from the stripe domain, resulting in the generation of skyrmions. The details were observed by employing Joule heating to overcome the pinning effect and manipulating the strength of the SOT to induce the branching and breaking of half skyrmions. These findings offer new insights into skyrmion generation and serve as an important step towards the development of highly efficient devices for processing and computing based on skyrmionics.