Final answer:
The autonomic nervous system's sympathetic division is responsible for transmitting messages from the spinal cord to the heart muscles and sweat glands, using a process involving preganglionic and postganglionic fibers and neurotransmitter release.
Step-by-step explanation:
Messages are transmitted from your spinal cord to your heart muscles and sweat glands by the autonomic nervous system, specifically the sympathetic division. The process begins with an axon from a neuron in the lateral horn of the spinal cord projecting to a sympathetic ganglion, where it synapses on a ganglionic neuron.
Following this, the preganglionic fibers release acetylcholine (ACh) which excites the ganglionic neuron through the nicotinic receptor. The postganglionic fibers then release neurotransmitters such as norepinephrine or, in the case of sweat glands, ACh onto their respective receptors, thereby effecting changes in the target organs.