Here is the trick:
Take the first digit (of a 2-digit number ending in 5),
or take the first two digits (of a three-digit number ending in 5),
multiply it by the next consecutive number, and place it in front of 25.

For example, 105 x 105 =
    step 1: (10 x 11) = 110
    step 2: 11025.

Why don't you use some algebra to prove that it works for 2-digit numbers?
Pick your favorite two-digit number that ends in 5.
Let's call it x5, where x could be 1, 2, 3, ..., 8, or 9.
Then x5 is really a shorthand notation for the integer represented by 10*x + 5.
When we square x5:
    (x5)^2 = (10*x + 5)^2 = 100*x^2 + 100*x + 25 =
    100(x^2 + x) + 25 = 100 * x * (x+1) + 25
    and that just happens to be the product of x and the next consecutive number (x + 1)
    with 25 placed at the end of it.
    Note that the 100 just moves the number over two digits (where the 25 is).