May've is the short spoken form of may have. The apostrophe stands where the ha of have has dropped out, so She may've gone home means She may have gone home: it is possible that she has already left. You hear it a lot in fast, informal speech, alongside could've, would've, should've, and might've. In careful writing, especially school work, teachers usually want may have written out in full. The meaning does not change when you shorten it; only the sound and the look on the page change. It is still the helping verb have, used after may to talk about a possible past action or a possible completed state.