I'm pretty sure "mai" is pronounced like "my", not "may". The recorded voice says it like "may". For example, the common phrase "e komo mai" in Hawaii is always pronounced like "my". An earlier recorded female voice in this lesson also pronounced mai like "may". What is correct? Mahalo!
I'm just learning, but I think it's best not to think in terms of exact translations to English words, but rather to concepts. Mai expresses several concepts. According to wehewehe.org the first meaning is "Directional part[icle], towards the speaker, this way" then gives examples like "Hele mai, come" and "Hāʻawi mai, give me." In all of these, the concept of "mai" is "toward the speaker." You see that also in the common expression "No hea mai 'oe?" = "Where do you come from?" In that expression, you're coming from some other place (no hea) to where the speaker is now (mai).