How do fibers control muscles?
Read "supply" as "carry action potentials to." When the action potential reaches the junction with the muscle (i.e., the neuromuscular junction), neurotransmitters are released into synapse. A similar membrane depolarization occurs on the muscle cell, ultimately leading to contraction.

Nerves visible to the naked eye are actually bundles of individual axons (hundreds, thousands, or tens of thousands of axons). Nerves have the appearance of branching because the individual axons that travel to muscle fibers don't all go to one muscle. 

For example the oculomotor nerve branches:



So what you see as the oculomotor nerve is really a collection of neurons. Some of these carry action potentials to the ciliary muscle and some to the sphincter pupillae. These are the parasympathetic components. There is a completely separate set of neurons (but still contained in the same nerve) that carry signals to the extraocular muscles.

This is really basic material that your professor should have explained to you.