The basis for stenography is that it records phonemes (sounds) instead of words. The stenographer learns to transcribe the sounds of language rather than the words, and turns those recorded sounds into actual words in a second phase of the process. That second phase occurs back in the office, outside the courtroom – what you see in the courtroom is just the first phase, wherein the sounds of the courtroom proceedings are recorded.
The machine that the stenographer uses is called a stenotype machine. You can think of it as a kind of specialized typewriter that records phonemes rather than letters and words. It is optimized for speed and reliability, as the court reporter must be able to keep up with even the fastest speaker, and you certainly don't want the machine to break down in the middle of proceedings.
One thing I learned at Xscribe is that court reporters make extensive use of shorthand “codes” – strokes (recorded phonemes) that don't actually reflect any sound heard in the courtroom, but rather are special symbols that only the court reporter knows that represent some piece of information. One of the most common uses for this is to make a short code that represents a long and complicated name. Some codes might be used by a particular court reporter in every one of their cases; others are made up on the fly for a particular case. These codes are always very personal to each particular court reporter. Much of what Xscribe's software did was to automate the process of turning these special codes into words, something that saved a vast amount of time for a court reporter. The loved Xscribe's products, and it was easy to see why.
These days, with the much more powerful computers that are available, the court reporting software just runs on ordinary PCs. I haven't talked with a court reporter for many years, but I'd bet they have trouble even imagining doing their job without a computer now...