You just can't divide a number by zero.
In fields, that is systems of mathematical objects, defined by certain rules, we can perform addition and multiplication. We have things called "inverses". Every field has an identity element, 1, and a zero element 0. Basically, multiplying by 1 does nothing, and adding 0 does nothing. Every non-zero element in the field has an inverse, i.e. for every "a" in the field there is a "b" such that ab = ba = 1. But 0 does not have an inverse.
An example of a field is the real numbers. When we "divide" real numbers together, we are really finding an inverse, then applying multiplication. Division is not its own thing, it is a shorthand for this process. x/y means find the inverse of y, then multiply x by that inverse. With 0, we fail at the first hurdle, as 0 does not have an inverse. There is no number, x for instance, such that 0 * x = 1. Division by 0 cannot be done. It is undefined.
Of course, we don't have to look at things through the field lens. But it's how this stuff is rigorously supported, so I think it's pertinent. Others have already stated more number-orientated intuitive ways of looking at this.