And also, why you would need to difference twice to remove quadratic trends [in a sequence of numbers].
(I was asked to add this as a question and answer, to give the answer more exposure from Why does differencing once remove not only linear but also nonlinear trends?)
The principle is based on the Method of Differences.
If you'll forgive some Python:
>>> import numpy as np
x>>> xs = np.arange(5)
>>> xs
array([0, 1, 2, 3, 4])
>>> ys_constant = 0.0 * xs + 1
>>> ys_constant
array([1., 1., 1., 1., 1.])
>>> np.diff(ys_constant)
array([0., 0., 0., 0.])
>>> ys_linear = 2.0 * xs + 1
>>> ys_linear
array([1., 3., 5., 7., 9.])
>>> np.diff(ys_linear)
array([2., 2., 2., 2.])
>>> ys_quad = xs**2 + 2.0*xs + 1
>>> ys_quad
array([ 1., 4., 9., 16., 25.])
>>> np.diff(ys_quad)
array([3., 5., 7., 9.])
# need the second difference to get constant behavior
>>> np.diff(np.diff(ys_quad))
array([2., 2., 2.])
>>> np.diff(ys_quad, n=2)
array([2., 2., 2.])
Now, that shows that the method works. But how/why? Consider the differences as simple approximations to a derivative $\frac{f(x)-f(x+\Delta)}{\Delta}$ where the $\Delta$ values is fixed at $1$ (so it disappears from the denominator and is use a "fixed increment" to the next input in the sequence, for example $x=2 \rightarrow x=3$).
Then, repeated differencing is like taking higher order derivatives. The first order derivative of a line (aka the slope of a line) is always constant. So, we only need a first order difference to remove a linear trend. The second order derivative of a quadratic likewise gets us to a constant (in $y=ax^2 + bx + c$ we throw away the $b,c$ and are left with $a$).