Curse of dimensionality
Suppose a p-dimensional hypercube. The edge length $e$ of the hypercube that captures fraction $r$ of the volume is $e = r^{1/p}$ because $r / 1 = e^p$.
For example, in 10-dimensions, edge length of 0.5 captures only 0.001 of the volume.
See (2.25). I will write the true value $f(x_0)$ as $f$ and prediction $\hat{y}$ as $y$. All expectations are across the training set $T$.
Variance is the expectation of squared deviation of a random variable from its expectation, $E [(y - E[y])^2]$. Bias is deviation of an estimator’s expectation from the true value, $E[y] - f$.
Then, $Var(y) + Bias(y)^2 = E [(y - E[y])^2] + (E[y] - f)^2$, which we expand to $E[y^2 - 2 y E[y] + E[y]^2] + (E[y]^2 - 2 E[y]f + f^2).$
By lineararity of expecation, this is equivalent to $E[y^2] - E[2 y E[y]] + E[E[y]^2] + E[y]^2 - 2 E[y]f + f^2 = E[y^2] - 2 E[y]^2 + E[y]^2 + E[y]^2 - 2 E[y]f + f^2,$ because the expectation $E[y]$ is not random (that is, $E[E[y]] = E[y]$). Terms cancel to $E[y^2] - 2 E[y]f + f^2$.
Mean Squared Error is $E[(f - y)^2] = E[f^2 - 2fy + y^2] = f^2 - 2fE[y] + E[y^2]$ because the true value $f$ is not random. Hence, we decomposed MSE into squared bias and variance.