Short answer is, because the 1/2" flexible tubing is short enough to not cause a major pressure loss. If you ran 1/2" line the whole way, it would be too much pressure drop.
Pressure drop across a pipeline is a function of all the friction losses added together. These include "major" losses (the official terminology, not mine) from the friction with the pipe itself, and "minor" losses from having to go through bends, tees, and valves.
The major losses are a function of the velocity (V) through the pipe (squared), the length (L) of the pipe, the diameter (D) of the pipe, the density (p) of the fluid, and the "roughness" (f) of the inside of the pipe. The diameter actually hits 5 times because reducing the diameter increases the velocity by the square of the reduction, which is then itself squared again, and the diameter comes in again in the friction equation. It looks something like this: 2*f*V^2*L/D
Minor losses are a function of the velocity squared, the density, and the specific character of the disturbance. 90 degree bends have more loss than 45s, long-radius bends have less than short, and so on.
So, increasing the pipe size greatly reduces the pressure drop in a pipeline when the gas is flowing because it slows the gas down and gives it less surface to have to rub up against.
The pressure drop because of friction is directly related to how long the pipe is and all of the pressure drops are additive. That means you can change one without affecting the others.
So, say you lose 1"WC for each foot of 1/2" pipe you go through, and 0.1"WC for each foot of 1" pipe you go through at a certain flow rate.
If you ran 50' with just 1" piping, the loss would be: 5"WC
If you ran 50' with just 1/2" piping, the loss would be: 50"WC
If you ran 47' with 1", and 3' with 1/2", the loss would be: 7.7"WC
7.7 is a lot lower than 50, so the point is, it's okay to go through a smaller section of pipe, as long as it's not too long. And certainly, going through a short piece of small pipe and then a long piece of large pipe is much better than going through a small pipe the whole way.
Best Answer
Propane appliances are not a problem, but the tanks are. Any propane cylinder or tank should be considered a potential hazard and not stored inside.
People often make exceptions for the small ~14 oz cylinders used for torches and portable stoves. Hardware stores will stock those small cylinders inside the store. However, tanks like the common 20 lb size used with gas BBQs aren't even allowed inside the store.
They're usually kept outside in steel or steel mesh cabinets that act as flame arrestors in addition to protecting the tanks. If you're talking about a 100 lb cylinder, that should definitely stay outside in a protected place away from anything flammable. Check out this advice on storage (tank position is also important).
You can pipe it into the residence, though, just as you would from a buried tank (the requirements are covered by code).
The gas, itself, determines the pressure. When the tank is closed, the pressure will be the same in any size tank. Some of the liquefied gas converts to gaseous form until the pressure is sufficient to keep the rest in liquid form. The size and shape of the tank might affect how quickly the tank can replace the gas being drawn off if you were feeding a monstrous requirement, but for typical residential use, appliances won't care what kind of tank is feeding it. You could use a 14 oz torch cylinder, but you would run out quickly.
I'm not familiar with the tank regulators and fittings that are required by code for different size tanks. Hopefully, someone else can address that aspect.