You cannot explain how something goes from a binary number to a video in 20 words or less!
If you are working in the decimal system, you need 10 numbers, 0-9. That is difficult to put in a machine. If you use the binary system you only need 0 and 1 which could be a switch being on or off, a memory blob being magnetized or not magnetized or electronically as ScottALot described.
There is no simple way for a machine to add say 10 + 3. It used to be done in complex mechanical adding machines. It can be done easily in binary, 1010 + 11 which added in binary give 1101 (8x1 + 4x1 + 2x0 + 1x1) or 13. By various electronic means you can make simple and complicated calculations. That is the basics of computing.
You can make a code such as ASCII so that if you tell a printer a binary number is text, it will convert it to a letter. ASCII 97 (1100001 in binary) if sent to a printer would be decoded as the letter "a". In reality it is far more complicated.
From there you can use binary to build applications do all sorts of things. Let's take another (over-simplified) example. A pixel located in your digital camera's CCD at coordinates row 25 column 732 sees an RGB colour 255,123,17 (shade of yellow) through the lens. You can send that to the picture file as
Row 11001
Column 1011011100
Red 11111111
Green 1111011
Blue 10001
All those values can be stored in memory as bits that are On (1) or Off (0). That stores the colour of just one pixel in the image. Your graphic application can take that data and recreate the pixel on your digital photo.
That's as far as I'm going to go. I hope it's of some help.