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Scilly.Suffolk
In the general context, torque is the (cross) product of the force applied and the moment arm. What this mean is if you have a large force applied far away from the pivot point, you have a large torque.
In an automobile, pushing on the gas a certain amount at a certain RPM will produce the same torque. By changing the moment arm, or the distance from the axis of rotation, you can change the force that is applied to the teeth (or the chain if you pretend the car’s transmission is like a bicycle, which is a fine assumption if you’re just curious and not serious about engineering cars) and if you change the moment arm going to the axle, you can change the torque applied to the axle. You can also change the force going from the tire to the road by changing the diameter of the tire, but that’s not really something you can change while you’re driving…
So, to recap, the engine outputs some torque. That torque can be scaled up or down to get more or less torque at the axle and thus the wheel. More torque means more force between to wheel and ground which means more acceleration (F = ma). Now the question is how does the torque get scaled up or down?
If you have a small gear at the engine and a big gear at the axle, then the moment arm at the engine is smaller than that at the axle while the force applied on the teeth is the same (due to newton’s third law). Since torque is force times moment arm, changing from a smaller moment arm to a bigger moment arm will increase torque, so a small gear at the engine turning a big gear at the axle will increase torque, but now consider that one rotation of the motor will result in less rotations of the axle because the axle’s gear is bigger. This is the tradeoff between torque and speed.
In first gear, essentially the gear at the engine is smallest and the gear at the axle is largest, to produce the most torque at the axle with the same gas flow into the engine.
In reality, the transmission in a car is more complicated to reduce the chance of failure, increase transmission lifetime, decrease size, etc. Real cars usually use some sort of planetary gear scheme, but the idea is the same where each “mode” or “gear” will have an equivalent gear sizing ratio between the engine and axle. First gear will have the smallest ratio meaning small engine gear to large axle gear. For the same engine output, you get more force at the wheels for greater acceleration.
The reason we don’t always drive in first gear is because of the speed tradeoff. The engine can only turn so fast, and the faster the wheels turn, the faster the engine has to turn. Upping the gear ratio allows the wheels to turn faster with the same engine speed, but you can’t accelerate as quickly or go up as steep hills.