How many times more than $0$? If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples? 
If you look at the graph $f(x)=\frac{10}{x}$, it shows that when $x$ approaches $0$, $f(x)$ approaches infinity. So it means I have infinity times more apples than you?
 A: It would be more accurate simply to say that the question has no answer -- your ten apples cannot be described as a multiple of my zero apples.
You can choose to call that "infinity times", but for most purposes that is just a way to fool yourself into thinking that you have answered the problem when in fact you haven't. More specifically, if you know you have "two times as many as 5" apples, you can use this to compute the exact number of apples -- but knowing that you have "infinity times as many as 0 apples" tells you nothing useful.
It's better just to leave the result of the division-by-zero undefined. You don't have any times zero apples.
A: You have to be careful because infinity ($\infty$) is not a number, so saying "I have infinity apples" doesn't really make sense.
It would be better to say "I have impossibly-many apples" perhaps.  And that should match your intuition from the example you gave: it is impossible to multiply zero by any number that would be large enough to make it equal to 10.
Zero is unique in this respect: any other number, no matter how big or small, can be multiplied by some other number to get 10.  But not zero.
