To 0/0 it means you need to find a unique number x such that 0*x=0
However any number can satisfy that thus x is not a unique number
According to one of my maths professors, since division is a binary operation (namely it takes two inputs and produce an output) we don't expect it to have multiple possible values (In fact it is even worse then similar multivalued things such as solving for sin x=0 or (exp(x))^y where x,y are complex numbers (where you would end up something like exp(x*y+n*pi) as the function exp(x) in the complex plane have the same values after you walking around it in a full circle) , in that it can take ANY possible numbers in the number system in question)
Therefore 0/0 we basically cannot tell anything useful from it, hence we cannot determine anything from it alone (indeterminate).
When you get to things like calculus and compare the behavior of functions such as f(x)/g(x) as x approach a certain value a, if you get f(a)/g(a)=0/0 it means you cannot tell anything other than the two functions will separately tend to zero.
In order to tell more, you need the L'Hopital Rule which said if f(a)/g(a) = 0/0 or infinity/infinity and f'(x)/g'(x) as x tend to a exist and is a finite value L, then similarly f(x)/g(x) as x tend to a will tend to L.
In layman terms, it compares the rate of f(x) relative to g(x). If f(x) change quicker than g(x) as x approaches a (I.e f'(x) > g'(x) as x approaches a) it means the final value of f(x)/g(x) as x approaches a (BUT NOT =a) is dictated by f(x). So if f(x) approaches zero near a then f(x)/g(x) will also approach zero near a.
So what about f(a)/g(a)=0/0? , in this case since you have plugged a value (number) into a function and it split out a number, then we should treat 0/0 like a number and the argument in the 1st paragraph applies, that is, we cannot have a unique or a discrete series of answers on what 0/0 is. (Even things like modular arithmetic e.g a mod 12 system like a clock, you still have something like x=x+12 but not something like x+a=all possible numbers)
So the ultimate conclusion is that because of how numbers and division in the usual sense are defined, 0/0 gives no useful result (having an expression such that it equals to any conceivable number is not very useful in solving problem, as you always end up all possible numbers as the answer, which in reality is not the case for nearly all problems)
Footnote: There is actually a number system where 0/0 and n/0 are defined, see below
https://en.wikipedia.org/wiki/Wheel_theory
It is very bizarre in that it has two types of division and 0/0+x=0/0
Still yet to get my head around it, let alone how it is used