From the intuitive standpoint, C is a contradiction because, supposing otherwise, truth cannot imply falsehood. Therefore, (I -> E) -> C is equivalent to ~ (I -> E), and this in turn implies I.1. (I>E)>C
2. C > ~C
More formally, Natural Deduction, is, in fact, natural, so here is an explanation in the natural language .
The proof can be broken into two parts. The first proves ~ (I -> E), and the second deduces I from that.
For the first part, assume I -> E. Then you get C with 1. Also, you get ~ C with 2. Obtaining C as before once more, you get a contradiction from C and ~ C. Therefore, the assumption I -> E was false and we have ~ (I -> E).
The second part is just a bit trickier because it uses the law of double negation. Assume ~ I and (separately) I. This, of course, gives a contradiction. Since everything follows from contradiction by ex falso quodlibet, we deduce E. Closing the assumption I, we get I -> E. (We still have ~ I open.) But, in the first step we deduced ~ (I -> E), so we get a contradiction again. Now, closing the the assumption ~ I we get ~~ I, from which by Double Negation we get I.