Explain the relationship between the binary numbering system and the function provided by the transistor.

The relation is the Voltage levels.

Explanation:

Basically the relation is related with voltages levels, that is, the computer has the main component called mother-board this component is the "brain" of the computing systems made of something called microprocessors, intel has core i7,i9 dual core, etc. The mother-board manage the information and save information in the memories, usually Random Access Memory (RAM) and hard disk, this information is understood by the computer as ones (1) and zeros(0), that means, that the computer process logic levels or binary numbers.

Since the beginning of computer developments was in this way because facilitates the process of computing,  and the transistor was an excellent candidate for doing this because the device is scalable, and consumed less power than vacuum tubes that was used in 70's, more or less. There are configurations of transistor that allow use a binary numeric system, such as NOT,OR and AND gates, these are the base of all computers, the computers are made of millions of this gates made with transistors, for example, the attached picture has a NOT gate, suppose that is wired to VDD volts and is refereed to ground GND or zero ( 0 )volts, if a voltage level close to GND is aplied in the input the output is VDD, and if a voltage close to VDD is applied to the input the output will be GND.

In other words the logic levels are read by the computer as ones and zeros but these are voltage levels that could be measured with voltmeters or specialized equipment and are produced by transistors configurations as presented in this answer.

LOGIC LEVELS WHERE THE VOLTAGE VDD IS 1 AND GND IS 0

INPUT:

             1   ______                         1      ______

        0___|             |_________0____|             |_____

OUTPUT

       1  ___              _______________              _____

       0        |______|                               |______|