David sent in this quite neat explanation:

The numbers that are diagonal to each other add up to make the same number because you're adding one that's lower or higher by 1, 2 or 3 to the number beside it.For example, in a 3x3 square, the number in the top right is 2 more than the number in the top left. The number in the bottom left is 2 less than the number in the bottom right. So when we add the top left and bottom right, and the top right and bottom left, we get the same total.

Esther who is 8 used letters to stand for the numbers (we call this algebra) and shows why the totals are the same:

I have found out that the sum of each diagonal pair in a square is always the same as the other in the same colour. We can write any square out like this:N | N+1 |

N+10 | N+11 |

If we add the diagonals we get 2N+11 each time.

For a 3x3 square the result was the same and the sum is twice the
number in the middle: N | N+2 | |

N+11 | ||

N+20 | N+22 |

The sum is 2N+22 which is 2 x (N+11).

The 4x4 square looks like this:N | N+3 | ||

N+11 | N+12 | ||

N+21 | N+22 | ||

N+30 | N+33 |

The sum of the opposite corners is 2N+33. The opposite corners of the small square in the middle also add up to the same number.

A 5x5 table looks like this:

N | N+4 | |||

N+22 | ||||

N+40 | N+44 |

The sum is 2N+44 which is twice the number in the
middle.

Thank you Esther, that's a very clear way of explaining.

Even if you haven't met algebra, you can still use ordinary words as David did. Well done!