What is the result of $\infty - \infty$? I would say $\infty - \infty=0$ because even though $\infty$ is an undetermined number, $\infty = \infty$. So $\infty-\infty=0$.
From a layman's perspective, imagine that I have an infinite number of hotel rooms, each numbered 1, 2, 3, 4, ... 
Then I give you all of them. I would have none left, so $\infty - \infty = 0$
On the other hand, if I give you all of the odd-numbered ones, then I still have an infinite number left. So $\infty - \infty = \infty$. 
Now suppose that I give you all of them except for the first seven. Then $\infty - \infty = 7$. 
While this doesn't explain why this is indeterminate, hopefully you can agree that it is indeterminate!
Infinity does not lead to contradiction, but we can not conceptualize $\infty$ as a number. The issue is similar to, what is $ + - \times$, where $-$ is the operator. The answer is undefined, because $+$ and $\times$ are not the kind of mathematical objects that $-$ acts upon. (We could of course define something, but then we are no longer talking about the usual operations, we are changing the meaning of the symbols.) 

We can not conceptualize $\infty$ as a number to a large part because $\infty - \infty$ does not have a good interpretations. In particular for any finite set, the size of the set, its cardinality, is the same as the place of its last member, its ordinal. So if I have a concrete set $\{a, 57, \text{apple}\}$, then one of the members is at the place $3$, because there are $3$ members of the set. That is no longer true for infinity. 

This is shown by the Ross–Littlewood paradox, you receive $10$ coins at each time step, and then you give a single coin back. The question is, how many coins do you have after infinite many steps. The problem is called a paradox because it has a simple and intuitive answer, unfortunately not many people agree which answer is simple and intuitive. 

The first answer is, you have $\infty$ coins, which is a notational shorthand for, the total number of coins in your possession growth without limit. Proponents of this answer parse the problem in terms of total value where individual coins loose their identity. One could interpret that as taking the ten coins to the bank, then wiring the value of a single coin back. The bank updates your balance and the operation at a time step $t$ is $b_{t+1}=b_{t} + 10 - 1$.

The other answer is $0$, because each coin only spends a finite time in your possession and so after an infinite amount of steps you gave each and every coin back. In this case, one puts the coin into a line, one by one, adds $10$ coins to the end of the line, and gives $1$ coin back from the front of the line. That means that the $n$-th coin spends the timesteps $n/10$ until $n$ in your possession, and you give it away in a finite amount of time. (You get the $65$-th coin at time step $6$ and give it away during the time step $65$.) In "the end," infinity does not have an end" you have given all coins away and so the balance is $0$.

The difference between the two scenarios is, in the second the order of the coins is preserved, in the first the coins are piled and one only looks at the total value of the coins. And if we modify the second scenario a little bit, we end up with different answers. If you give back always the last coin you received, then you keep coins  $1,\dots,9$ from the first step, coins $11, \dots, 19$ from the second and so on and you get to keep an infinite amount. (I use the order of received here to number the coins, not the place the coins lie in at the end.) If you take the ten coins, put $1$ on a pile, and you put $9$ in a line, and you do that for the first $n$ steps and after that you get lazy and just put all the coins into the line, then you will give all the coins in the line away, but you end up with a pile of $n$ coins. 

These different procedures could all be plausibly called $\infty - \infty$, and we just constructed ways to parse that expression in a way to get any answer between $0$ and $\infty$. This is the issue with infinity, we have to be precise how to conceptualize $\infty$ and we can not just treat $\infty$ like a kind of number.