How does a stimulus cause threshold potential across a neuron to happen?

Acetylcholine or a neurotransmitter depolarizes the membrane

A stimulus causes "small" sodium channels to open slightly,
allowing sodium ions to "dribble" into the neuron.  

When the indribbling sodium ions make the inside positive enough to reach the threshold potential, the voltage-regulated sodium channels open, sodium ions "rush" in, and the impulse begins.

I'm not sure this is precisely true for all other kinds of neurons, but this holds for cholinergic neurons.

The presynaptic neuron releases acetylcholine into the synapse when an action potential reaches the axon terminal. This acetylcholine binds to acetycholine receptors on the postsynaptic neuron.

The acetylcholine receptors are actually ligand-gated sodium channels. That is, when acetylcholine binds, the channel opens and sodium diffuses in down its concentration gradient and slightly depolarizes the neuron. Only when there is enough acetylcholine released into the synapse (i.e. the stimulus is strong enough) does the membrane potential reach threshold, at which point the voltage-gated sodium channels open and sodium floods in, leading to the characteristic depolarization of an action potential.