The answer to this is really straightforward: Monocots have plant parts in threes or multiples of threes because that is part of taxonomic definition of monocots.
More detail is probably what you are hoping for, though. Angiosperms (flowering plants) are divided (tentatively though because of some indistinguishable features in some plants) into two major classes, the Dicots and the Monocots. Yet some plants have "fuzzy" characteristics, like water lilies that appear to have a single lobed monocot cotyledon that may actually have been a double lobed cotyledon that fused into a single lobed one over time.
Features that generally identify monocots are these:
- single cotyledons within the embryo
- singly grouped pollen as furrow or pore
- flower parts in threes or multiples of threes
- parallel leaf veins, not reticulated leaf veins
- scattered stem vascular bundles
- random, adventitious, roots, not a radicle root from the embryo
- absence of secondary growth (University of California Berkeley)