The inner loop takes an array and an index i in the array, and rearranges the array to put the largest value at index i. This is done in such a way that if the sub-array up to but not including i is sorted, then afterwords the sort will be extended such that the sub-array up to and including i will be sorted.
The outer loop says, just call in to the inner loop with i from 1 to n. This will result in a sorted array, because the initial array of a single element is trivially sorted, and then we'll have by the sort-extension property of the inner loop that the progressively larger sub-arrays of 2, 3, ... up to N elements become sorted.
You can see that the inner loop has the sort extension property by thinking about two cases.
Case 1: If the i-th element is already larger than all previous elements, it won't touch any of the elements before i, and will only potentially make the i-th element bigger, thus clearly extending the sort though the i-th element.
Case 2: If the i-th element is smaller than one of the previous elements, it won't touch any elements until reaching that smaller element; then it will start a series of swaps until it reaches index i, at which point the the original value of the i-th elment is left inserted in sorted position into the sub-array up to and including i. So at this moment we have the sort extension property. From there, it may increase the value of the i-th element and change later elements, but of both those will preserve the sub-array sort.