At many chemical synapses, an arriving action potential triggers neurotransmitter release by opening voltage-gated Ca$^{2+}$ channels in the presynaptic terminal.

The sequence is

action potential depolarizes terminal
        ↓
voltage-gated Ca2+ channels open
        ↓
Ca2+ enters down its electrochemical gradient
        ↓
Ca2+-sensing release machinery is activated
        ↓
synaptic vesicle fuses with plasma membrane
        ↓
neurotransmitter is released

Why calcium is an effective trigger

Resting cytosolic free Ca$^{2+}$ is kept low. Extracellular Ca$^{2+}$ is much higher, and the negative interior of the cell also favors entry of positive Ca$^{2+}$ ions.

Opening presynaptic Ca$^{2+}$ channels therefore produces a rapid local rise in Ca$^{2+}$ near the release sites.

Ca$^{2+}$ binds proteins in the vesicle-fusion machinery, increasing the probability that a docked vesicle will fuse with the presynaptic membrane.

Release is quantized by vesicles

A synaptic vesicle contains a finite packet of neurotransmitter. Fusion releases approximately one vesicle's packet into the synaptic cleft. Stronger presynaptic Ca$^{2+}$ signals generally increase the probability and number of vesicles released rather than continuously enlarging one indivisible packet.

Vesicle membrane is recycled

Exocytosis adds vesicle membrane to the plasma membrane. Membrane and vesicle proteins are subsequently retrieved and reused through endocytic and trafficking pathways, helping maintain the presynaptic terminal during repeated signaling.

The important conversion is therefore voltage → Ca$^{2+}$ entry → regulated exocytosis. Calcium does not carry the neurotransmitter across the membrane; it acts as the intracellular trigger for vesicle fusion.