Well, step back from astronomy for a minute. When you go out and buy a new printer, you can be virtually certain that it will work with all of the programs on your computer. Likewise, when you install a new program on your computer, you can be virtually certain that it can print to your existing printer, even if it's no longer in production. We take this for granted. Printers come with a disk that installs the driver for the printer. The driver takes care of all of the details for that particular printer, leaving all of your Windows programs with a common printer-agnostic way to send pages to paper.
OK, back to astronomy. Until ASCOM, each astronomy program that needed to control telescopes, focusers, and so forth had to include its own code for all of the different instrument types out there. Keeping up with new instruments, supporting old ones, and dealing with firmware revisions is a tremendous burden. Every astronomy software developer is faced with (re)writing code for every device he intends to support. Furthermore, astronomy device manufacturers are faced with having to beg an array of astronomy software vendors to support their device in the future, delaying adoption of their new devices.
ASCOM eliminates these problems. Most programs that need to control telescopes, focusers, etc., now expect a driver to be available for those instruments. For example, you may have several programs that need to control your telescope (planetarium, imaging software, alignment assistance tool). If there is a driver for your mount, you can be virtually certain that all of these programs can control it. To find out, you ask your mount maker "Does your mount have an ASCOM driver?" If so, you're all set. No more asking a bunch of software developers "Does your software support my mount?"
For more information, see How Does ASCOM Work?