What is the purpose of showing some numbers exist? For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?

You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number.
If the real numbers didn't have a number whose square is 2, that would be a rather serious defect; it would mean that the real numbers cannot be used as the setting for the kinds of mathematics where one wants to take a square root of $2$.
Basically, things cut both ways; while you could use the argument as justifying the idea of taking the square root of 2 is a useful notion, the more important aspect is that it justifies the idea of the real numbers is a useful notion.

Also, the argument is useful as a demonstrates of how to use the completeness properties to prove things.
Furthermore, it reinforces the notion that this type of reasoning can be used to define specific things. Some people have a lot of difficulty with this type of argument; e.g. "you haven't defined $\sqrt{2}$, you've just defined a way to get arbitrarily close to a square root of 2 without ever reaching it". It's a lot easier to dispel such misconceptions when the subject is something as clearly understood as $\sqrt{2}$.
