The image says it all, but let’s be sure we understand it all, because this is an important one.
Pretend you’re looking through a pair of binoculars with about a 7 power magnification at a bird in your front yard. It jiggles around a little bit as you try to hold it still, but you can make out the bird well enough, and the jiggling isn’t too bad. Now imagine that the bird flies down the street and lands in another tree there. You bring out a telescope with a 200 power magnification and try to hold it on the bird. You’re disappointed to find that without resting the telescope on some solid object, there’s no way you can hold it still enough to see the bird. The reason for that is because when you increased the magnification of the bird, you also increased the magnification of your movements and shaking. What can you do? Either get the telescope on a tripod, or find some way to cut “a thinner slice of time”.
Lens length is how magnification is measured in photography lenses. The longer the lens: the greater the magnification. Therefore the longer the lens, the thinner the slice of time that’s needed to eliminate motion blurring… thinner slice of time of course means faster shutter speed.
A “reciprocal” in math is the number that when multiplied with the orginal number equals one. So if you’re shooting with a 300mm lens, your MINIMUM shutter speed should be 1/300th of a second. For a 500mm lens it should be not slower than 1/500th, and so forth.
And by the way, about the “never slower than 1/60th”? Okay, but why not hold yourself to a higher standard and say “never slower than 1/100th”?