At their simplest, every program on a computer is made up of a series of single instructions to the computer's CPU that tell it what to do. Think of it as walking someone through every single, agonizing step to do even the simplest of processes. Because of this limitation of computers, creating programs for the computer used to be a very difficult and painstaking job.
Luckily, that all changed. With the development of computer compilers, programmers were finally able to write single lines of code that, when translated, would be the equivalent of thousands of lines of single instructions to the CPU. Computer libraries are essential to the working of these compilers in doing their job.
Have you ever been to a new country and tried to speak the language? If you have, it is likely that you kept a phrasebook handy. This is a useful way to say new phrases without forcing yourself to look up each individual word in a dictionary to make your phrase. This is similar to what computing libraries provide for the computer programmer. Rather than having them 'reinvent the wheel' every time they need to do something, computing libraries are the 'phrasebook' that makes their lives simpler.
Whenever a programmer needs to perform a function, he or she can use some of the many computing libraries that exist to do the work. While making these functions from scratch might have taken the computer programmer hours, days, or even longer, he or she can call a function through the computing libraries in often as little as a single line of code.
Without the development of computing libraries, it is difficult to imagine the world as we know it. Computing libraries have enabled the development of complex pieces of software in reasonably quick time spans that otherwise might have taken much, much longer.