Accreting millisecond X-ray pulsars (AMXPs) are an important subclass of low-mass X-ray binaries (LMXBs), in which coherent millisecond X-ray pulsations can be observed during outburst states. They have dual characteristics of LMXBs and millisecond pulsars, providing a direct confirmation for the recycling scenario. However, their formation is not well understood. In this work, we simulate the evolution of LMXBs with the MESA code to explore the formation and evolution of AMXPs. Based on the binary evolutionary model of LMXBs and the model of accretion disk instability, we find that most of the observed AMXPs can be produced from LMXBs with orbital periods at the onset of Roche lobe overflow close to the bifurcation period and their observed properties can be explained by our models. The AMXPs with main sequence (MS) donors ultimately evolve into AMXPs with extremely low-mass He white dwarf donors. Moreover, our results indicate that these AMXPs with MS donors are likely to have donor stars near the terminal-age main sequence.