Here is their example coded in R: a sample size $1000$ with points in the $p$-dimensional hypercube $[-1,1]^p$ with $p=10$:
set.seed(1)
p <- 10
samples <- 1000
matdat <- matrix(runif(samples*p, min = -1, max = 1), ncol=p)
Y <- exp(-8*rowSums(matdat^2))
and you get for the nearest neighbour
> max(Y)
[1] 0.005093686
rather than $1$ at the origin.
Some of this is accentuated by their particular function $Y=f(X)$, but the thousand points must be a distance between $0$ and $\sqrt{p}$ (here about $3.162$) from the origin. In this particular sample the nearest is in fact $0.812$ away and the furthest $2.534$ from the origin, and these are not unusual examples
In general, the distances of points in this hypercube from the origin are approximately normally distributed with mean about $\sqrt{\frac{p}3 -\frac1{15}}$ and standard deviation about $\sqrt{\frac1{15}}\approx 0.258$. It is this small standard deviation not changing materially with $p$ which gives rise to this particular form of the curse of dimensionality
If you were to increase $p$ to $1000$ then you would expect more than $99.999\%$ of points to be at distances between $17$ and $19.5$ from the origin, even when they could have been anywhere between $0$ and $31.62$ away. So not only would the nearest neighbours probably be more than half the maximum distance away, but the furthest sample points would not be much further away than the nearest ones, making the whole predictive process highly dubious at high dimensions