How do you know what parts to use in a circuit?
There are a few parts to this:

When a designer is designing, they first need to come up with an architecture (say, low noise amplifier (LNA) - filter - power amplifier - antenna)
They then need to consider the required specifications for the building blocks. How much gain do we need in our LNA? What is the maximum noise? What is our bandwidth? Are we going to use differential or single ended circuits? How sharp does our filter roll-off need to be? These all follow from system requirements and such. 

This is a very, very important step, and according to some this step is what can differentiate hobby electronics from professional electronics, but that is another story.

How do you come up with this architecture and the specifications? Understanding of the mathematical principles and signal theory. For example, in your example of the RC car controller, understanding how to mix different signals to generate the RF signal, how to manipulate this signal, etc...

Once we have specifications, we need to look at circuits we can use to get to them. For amplifiers, there are a few ones. For example, a common-source amplifier:







simulate this circuit &ndash; Schematic created using CircuitLab

Or maybe a invertor-style design:







simulate this circuit

There are many more, and each has their advantages and disadvantages. As a designer, you learn what different building blocks look like and what they are good/bad at. 

You can then look at the circuit and see what impact different parameters have. For example, the DC input voltage changes the operating point of the transistors, which influences gain, noise, bandwidth, etc. The same goes for the current source in the top example - this can be built using just a resistor to Vsupply, and the value of this resistor has an impact on the behavior. What this impact is can be derived from the equations. 

The designer has to now play with the values of different components and try and meet the specifications. Perhaps they will conclude that the chosen topology is not suitable after all, and they need to go for a different topology. 

All in all, there is no magic knowledge that designers have to come up with these values and circuits - it is knowledge and theory that they studied and learned that allows them to design these circuits.