Why a car needs more than one gear
Why an engine only makes useful power over a narrow range, and what gears do about it.
A car needs more than one gear because an engine only produces strong, usable power across a narrow band of revolutions, and a gearbox exists purely to keep the engine inside that band while the car's actual road speed changes far more widely underneath it.
I found a copy of Carroll Smith's Tune to Win, from 1978, on a shelf in the team's workshop in my first week, wedged between service manuals nobody had opened in years, and it was the first book I read that treated a racing car as a chain of cause and effect rather than a set of things everyone just accepted.
Why an engine only pulls hard in the middle of its range
An engine's crankshaft can physically spin across a huge range, from a slow idle up to a screaming limit near the point where valves and pistons risk damage. Within that whole range, though, the engine only turns strong torque into the crankshaft over a much narrower window, usually somewhere in the middle to upper part of the rev range. Below that window the engine is turning too slowly for each intake stroke to fill the cylinder properly, and above it the engine is spinning too fast for the intake and exhaust events to keep up, so torque falls away at both ends. The valves, the intake tract and the exhaust system are all tuned to work together at a particular speed of gas moving through them, which is why an engine feels strong and eager in the middle of its range and flat at either end of it. A gearbox's whole job is to let the driver choose how the engine's narrow, torque-rich window gets translated into road speed, because the road speed a car needs to travel at spans a far wider range than the engine's own comfortable range does.
A two-to-one engine on a ten-to-one road
A typical engine's usable torque band might only cover a two-to-one spread of revolutions, from perhaps half its peak speed up to its limit, and yet the same car has to cover something closer to a ten-to-one spread of road speeds between walking pace and its top speed. A single gear cannot bridge that gap, since fixing the ratio between engine speed and road speed for a two-to-one engine range would only ever produce a two-to-one road speed range. Stacking several gears, each covering its own slice of road speed while keeping the engine inside its narrow productive window, is what lets the same two-to-one engine multiply out into the much wider spread of speeds the car is actually asked to reach. A car with five or six gears is carrying five or six overlapping copies of that narrow window, each shifted to cover a different slice of road speed.
Legs on a geared bicycle
Anyone who has ridden a bicycle with gears has already felt this exact problem from the other side. Pedalling down a steep hill in too low a gear, the pedals spin so fast that the legs are just flailing, unable to add anything useful to the bike's speed no matter how quickly they move. Grinding up a steep hill in too high a gear does the opposite: the pedals barely turn, the legs strain against a load that is too great for the muscles to push through at a reasonable cadence, and the bike nearly stalls. Between those two failures sits a comfortable pedalling speed where the legs can push hard and keep pushing, and a good rider changes gear constantly to keep the legs spinning near that comfortable rate whatever the hill is doing. A car's engine has an equivalent comfortable rate, and the gearbox keeps it there the same way the bicycle's chain and sprockets do, stretching one narrow, efficient range of motion to cover whatever the terrain, or in the car's case the driver's right foot, is demanding at that moment.
Close ratios for racing, and gearing for each circuit
This is why a racing gearbox looks so different from a road car's. A road car's gears are spaced to make the most of a very wide, forgiving engine, so the jumps between gears can be large without the engine ever falling badly out of its usable range. A racing engine's usable torque band is narrower and set higher in the rev range, tuned for maximum output rather than flexibility, so the gaps between gears have to be much closer together to avoid the engine dropping out of that narrow band the instant it changes gear. That is also why choosing gear ratios is treated as a genuine part of setting a car up for a specific circuit, since a track with several tight, slow corners needs the gearbox to keep the engine in its power band coming out of every one of them, while a track with long straights needs a top gear tall enough not to bump against the engine's limit before the straight runs out. A single gear set wrong costs time on every lap: too tall out of a corner and the engine labours below its best range on the run to the next braking point, too short and it forces an early shift right where the driver would rather keep accelerating.