It's hard to explain something when it seem pretty clear to you, maybe with some examples you'll get it too:

Consecutive numbers:
1 and 2, 2 and 3, 3 and 4, etc.

Not consecutive:
1 and 3, 2 and 4, 3 and 5, etc.

By not consecutive we mean not exactly 1 larger than the previous element of the array.

What does this statement mean?

The first non-consecutive number is 7. Not a kata issue.

By not consecutive we mean not exactly 1 larger than the previous element of the array.

Exactly

Testing for first_non_consecutive([5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45])

Here as well, there seems an issue.

let: Array name is arr

arr[1]-arr[0] = 2
So, when difference becomes other than 2, the non-consective number is next number.

arr[2]-arr[1] = 1, so first non-consecutive number should be 8, not 7