The low thermal conductivity of phase change materials restricts their application fields such as thermal storage and electronic equipment cooling. In order to enhance the heat charging capacity of the phase change unit, fractal fins inspired by plant leaves were designed and manufactured. The changes in the solid–liquid interface, temperature distribution and liquid fraction in the phase change units with fractal fins during melting were investigated experimentally and compared units with the conventional rectangular fin. The results show that fractal fins have better heat transfer enhancement effects than rectangular fins because the enhancement of heat conduction exceeds the suppression of natural convection. Increasing the number of fins can also shorten the melting time and make the temperature distribution more uniform. Compared with the one rectangular fin unit, the full melting time of the unit with three fractal fins is reduced by 17.07%, and the bottom surface temperature is reduced by 27.47%. However, increasing the number of fins while using tree-like fractal fins may cause the fins to inhibit natural convection more than enhance heat conduction. The research in this paper will provide a better understanding of the melting process of phase change units with fins and provide data for future numerical simulations.