If you've read How Hearing Works, you know that sound is essentially a disturbance in air pressure. When you strum a guitar string, for instance, that string vibrates (just as our vocal chords do when we speak). Those vibrations cause fluctuations in air pressure as they move outward, traveling in waves. When these waves of pressure variations reach our eardrums, our eardrums vibrate, and our brains interpret those vibrations as noise.
Our eardrums, like the larger side of a stethoscope's chestpiece, are diaphragms.
When a doctor or nurse places a stethoscope diaphragm on a patient's chest, sound waves traveling through the patient's body cause the flat surface of the diaphragm to vibrate. Those vibrations would travel outward if the the diaphragm were a standalone device, but because the vibrating object is attached to a tube, the sound waves are channeled in a specific direction.
Each wave bounces, or reflects, off the inside walls of the rubber tube, a process called multiple reflection. In this way, each wave, in succession, reaches the eartips, or rubber nubs on the ends of the device, and finally the listener's eardrums.
The waves of high-pitch sounds, like breath and heartbeats, are traveling at higher frequencies, meaning they cause a greater number of pressure fluctuations in a given time period. Higher-pitch sounds will directly vibrate the surface area of the large, flat disc (and the plastic disc inside). This basically means the sound waves caused by the opening and closing of an artery, for instance, are the same ones that travel through the stethoscope tubing to the listener's ears.
The bell works somewhat differently. Rather than picking up the vibrations caused by the artery's movement directly, it picks up the vibrations in the skin caused by that movement. The smaller, hollow bell contacts the patient with less surface area -- just the thin, metal rim. Lower-pitch sounds, which may have a harder time vibrating the large diaphragm, still vibrate the skin as they move outward. The skin then vibrates the bell.
Because the vibrations hitting the chestpiece are funneled into a narrow tube, instead of being allowed to travel outward at will, more of them reach the eardrum. In this way, the sounds they're carrying are amplified.
It's a neat trick. Using a stethoscope, a person more than 2 feet (0.6 meters) away from a patient's chest can hear louder heart sounds than a person whose ear is in direct contact with the patient. Diagnostically, this makes the stethoscope an invaluable medical tool.
Olfactorally, it makes it a godsend, just in case some patients today are still practicing hygiene at the early 19th-century-era standard. Sometimes, even in medicine, distance is a good thing.