What does it mean when we say most of the points in a hypercube are at the boundary?
Speaking of '$99\%$ of the points in a hypercube' is a bit misleading since a hypercube contains infinitely many points. Let's talk about volume instead.

The volume of a hypercube is the product of its side lengths.
For the 50-dimensional unit hypercube we get $$\text{Total volume} = \underbrace{1 \times 1 \times \dots \times 1}_{50 \text{ times}} = 1^{50} = 1.$$

Now let us exclude the boundaries of the hypercube and look at the 'interior' (I put this in quotation marks because the mathematical term interior has a very different meaning). We only keep the points $x = (x_1, x_2, \dots, x_{50})$ that satisfy
$$
0.05 &lt; x_1 &lt; 0.95 \,\text{ and }\, 0.05 &lt; x_2 &lt; 0.95 \,\text{ and }\, \dots 
\,\text{ and }\, 0.05 &lt; x_{50} &lt; 0.95.
$$
What is the volume of this 'interior'? Well, the 'interior' is again a hypercube, and the length of each side is $0.9$ ($=0.95 - 0.05$ ... it helps to imagine this in two and three dimensions).
So the volume is $$\text{Interior volume} = \underbrace{0.9 \times 0.9 \times \dots \times 0.9}_{50 \text{ times}} = 0.9^{50} \approx 0.005.$$
Conclude that the volume of the 'boundary' (defined as the unit hypercube without the 'interior') is $1 - 0.9^{50} \approx 0.995.$

This shows that $99.5\%$ of the volume of a 50-dimensional hypercube is concentrated on its 'boundary'.



Follow-up: ignatius raised an interesting question on how this is connected to probability. Here is an example.

Say you came up with a (machine learning) model that predicts housing prices based on 50 input parameters. All 50 input parameters are independent and uniformly distributed between $0$ and $1$.

Let us say that your model works very well if none of the input parameters is extreme: As long as every input parameter stays between $0.05$ and $0.95$, your model predicts the housing price almost perfectly.
But if one or more input parameters are extreme (smaller than $0.05$ or larger than $0.95$), the predictions of your model are absolutely terrible.

Any given input parameter is extreme with a probability of only $10\%$. So clearly this is a good model, right?
No! The probability that at least one of the $50$ parameters is extreme is $1 - 0.9^{50} \approx 0.995.$
So in $99.5\%$ of the cases, your model's prediction is terrible.

Rule of thumb: In high dimensions, extreme observations are the rule and not the exception.