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I asked one of the maths teachers at the college I work at and he gave the following explanation:

The trick relies on the fact that if you take any number, re-arrange the digits and subtract, the answer is always a multiple of 9. For example 43 - 34 =9 or 92 - 29 = 63. If you try 3-digit numbers, the same thing works: 351 - 153 = 198.

The sum of the digits of the answer is also a multiple of 9 (6+3 =9 1+9+8 = 1.

So we take any 4-digit number (say 8321) and rearrange it to 2381 (say). Now subtract to get 5940. We know the sum of the digits is a multiple of 9 (5+9+4+0 =1. So if I know three of them I just subtract the sum of the three from the next multiple of 9 to find the fourth. (suppose I am given 9, 4 and 0, then I add them up to get 13, and subtract 13 from the next highest multiple of 9 (1 to get the missing number 5.

So it should work every time, if you do your subtraction right!

The answer!!

The result must be a multiple of 9, the sum of its digits must
also be a multiple of 9. Now you are given all the digits but one,
so you add all the digits but one, and that missing digit must be
just enough to add to the sum of the other digits to make a multiple
of 9.

The Maths:
Every positive integer is a multiple of 9 plus the sum of its
digits.

Here is why. Take 7843. That's equal to

7×1000 + 8×100 + 4×10 + 3 which equals

7×(999 + 1) + 8×(99 + 1) + 4×(9 + 1) + 3 which equals

7×999 + 7×1 + 8×99 + 8×1 + 4×9 + 4×1 + 3 which equals

7×999 + 7 + 8×99 + 8 + 4×9 + 4 + 3 which equals

(7×999 + 8×99 + 4×9) + (7 + 8 + 4 + 3) which equals

9(7×111 + 8×11 + 4) + (7 + 8 + 4 + 3) which equals

a multiple of 9 + the sum of the digits.

Now when the number is rearranged, that number too is a multiple of 9
plus the sum of its digits.

Since the sum of the digits is the same no matter how the digits are
rearranged, when you subtract you get a multiple of 9 because the sum
of the digits cancel and leave the difference of two multiples of 9,
which is a multiple of 9.

Unfortunately I didn't work it out myself, but was intrigued and google came to the rescue!!

The trick relies on the fact that if you take any number, re-arrange the digits and subtract, the answer is always a multiple of 9. For example 43 - 34 =9 or 92 - 29 = 63. If you try 3-digit numbers, the same thing works: 351 - 153 = 198.

The sum of the digits of the answer is also a multiple of 9 (6+3 =9 1+9+8 = 18 ).

So we take any 4-digit number (say 8321) and rearrange it to 2381 (say). Now subtract to get 5940. We know the sum of the digits is a multiple of 9 (5+9+4+0 =18 ). So if I know three of them I just subtract the sum of the three from the next multiple of 9 to find the fourth. (suppose I am given 9, 4 and 0, then I add them up to get 13, and subtract 13 from the next highest multiple of 9 (18 ) to get the missing number 5.