Evaporation is the strongest candidate to explain the Mpemba effect. As hot water placed in an open container begins to cool, the overall mass decreases as some of the water evaporates. With less water to freeze, the process can take less time. But this doesn稚 always work, especially when using closed containers that prevent evaporated water from escaping.
And evaporation may not be the only reason the water can freeze more quickly. There may be less dissolved gas in the warmer water, which can reduce its ability to conduct heat, allowing it to cool faster. However, Polish physicists in the 1980s were unable to conclusively demonstrate this relationship.
My explanation for this effect is that most of these experiments seem to be conducted in refrigerators. What I suspect happens, is that hot water will warm the ambient air inside the freezer much quicker than cold water, so that the thermostat and compressor kick on immediately. Whereas the cold water takes another 5-10 minutes or more to register a temperature difference in the thermostat to kick on the compressor. Once the compressor kicks on, it keeps running until it reaches the cold side of the thermostat's "window" of temperature.