Mark Watney has a problem. Not only has he been stranded on Mars, but he has no way of communicating that problem with his former crew or mission control back on Earth. The storm destroyed the main communications dish, and the other three backup communication systems are located on the Mars Ascent Vehicle which the crew used to escape from the Martian storm.
Watney's solution is to dig up the Pathfinder rover and fix it up so that the camera is able to show him and communicate with Earth. The big problem is that the camera is essentially a one-way communication tool. Watney can write messages on paper and hold them up to the camera; however, he is also limited by the amount of paper available to him. NASA is capable of moving the camera in a 360 degree rotation, so NASA can move it one way for "yes" and the other way for "no."
The problem is the English alphabet. It has too many characters, and this is where the movie fails in comparison to the book. In the book, readers are walked through Watney's thought process about how to use the camera to communicate. One thought is to put each letter of the alphabet around the camera, but that means the camera has less than 14 degrees of rotation between letters. It's too easy to make a mistake with. Watney needs fewer points on the circle, but he needs just as many characters. The hexadecimal system gives him that option.
The hexadecimal system has sixteen total characters: 0–9 and A–F. The real genius is that Watney combines the hexadecimal system with an ASCII table. Number and letter combinations correspond to 255 individual characters. This means that 44 corresponds to a specific letter, number, or symbol and 4D corresponds to a different letter, number, or symbol. When NASA moves the camera to point out 48 4F 57 41 4C 49 56 45, Watney can then translate that using the ASCII table to read "How Alive."