The key is understanding how divisions between 0 and 1 work. Say you take 2/0.5, you end up with 4. 2/0.25 you end up with 8. As you can see, those numbers get big fast. 1/0.0001 is 10,000.

As you approach 0, you get increasingly large numbers. If you flip it negative, again as you approach -0, you have increasingly big negative numbers. As you approach 0 from both sides, you approach positive and negative infinity. But what goes in the middle, at exactly zero? We don't know, There's no sensible value there, so it's considered to be undefined.

In computers, it's usually either an error, or represented with NaN (Not a Number) when you want to avoid throwing an error condition. NaN is defined so that any operations involving NaN is also NaN, so your entire equation becomes NaN.

Easiest way to visualize that is to input y = 1/x in a graph calculator (Desmos is nice). You clearly see it going to negative infinity then back at positive infinity.

With some other operations like negative square roots, we've made up the imaginary number i, which is defined to be the square root of -1, and we can make it do useful things. But what can we reasonably do with the result of dividing by zero? It behaves like infinities: you can't really add or multiply them, or divide them. You're just stuck with it. It's impossible to represent, it's Not a Number.