Intuition/informal argument:
how is this valid?
Let's start by reviewing the finite days.
On day $0$, we take no numbers, put them in the left and right sets, and have $0:=\{ \,\mid\,\}$.
On day $1$, we take that number, put it in the left or right set ($\{ 0\mid0\}$ would break the inequality numbers are supposed to satisfy), and get $1:=\{ 0\mid\,\}$ and $-1:=\{ \,\mid0\}$.
On day $2$, we take the numbers we have so far, and put some of them in a pair sets in all legal ways that obey the inequality numbers are supposed to satisfy (so $\{ 1\mid-1\}$ is not allowed), and get new numbers like $\frac{1}{2}:=\{ 0\mid1\} =\{ -1,0\mid1\}$ , and things equal to old numbers like $\{ -1\mid1\} =0$.
In general, on day $n$ for finite $n$, we take all the numbers so far, and put them in sets in all legal ways.
Now, just call $F$ the set of all the numbers we could get on some finite day. Then we can take all the numbers in $F$, and put them in sets in legal ways, getting new numbers like $\{ 0,1,2,\ldots\mid\,\}$ and $\left\{ 0\mid1,\frac{1}{2},\frac{1}{4},\dots\right\}$ .
We can then do the "next day" step again, to build new numbers like $\left\{ \{ 0,1,2,\ldots\mid\,\} \mid\,\right\}$ .
For historical reasons dating back to Cantor, it's conventional to use the Greek letter $\omega$ to denote both the step we did with $F$ ("day $\omega$") and the surreal number $\{ 0,1,2,\ldots\mid\,\}$ .
Axiomatic/Formal arguments
What axiom allows us to talk about "what happens after infinite days"?
This depends very precisely on what you meant. Throughout, I'll assume you want to deal with standard ZF. Different books have slightly different statements of axioms, but I'll sketch the main points.
Building an Infinite Set
If you meant something like "how can we build a set like $\{ 0,1,2,\ldots\}$?", then the main axiom is the axiom of infinity, which basically declares the existence of a superset of $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\ldots\}$. Then, as Wikipedia sketches, we can then use things like (restricted) comprehension to get down to exactly that set. Under von Neumann's convention, the elements of that set would be labeled $0,1,2,\dots$, so that we would have a set $\{0,1,2,\ldots\}$. This set is the ordinal (not the surreal number) known as $\omega$.
The Day After
If you meant something like "how can we have a set to represent 'day $\omega+1$'?", then we can build on the previous construction. Once we have $\omega$, then we can use pairing to form $\{\omega,\omega\}=\{\omega\}$. Then we can use it again to form $\{\omega,\{\omega\}\}$. Finally, we can use union to make (at least a set containing) $\bigcup\{\omega,\{\omega\}\}=\{\omega,0,1,2,\ldots\}$. Traditionally, and generalized by ordinal addition, this set is called the ordinal (not the surreal number) $\omega+1$.
Building Surreals
But to actually and truly build a surreal like $\{0,1,2,\ldots\mid\,\}$, we need something that lets us write down that left set of surreals. We want to write something like $\{n^\text{th}\text{ nonnegative surreal with empty right set}\mid n\in\mathbb{N}\}$. We can use the ordinal $\omega$ in place of $\mathbb N$, and write down a complicated logical formula for $n^\text{th}\text{ nonnegative surreal with empty right set}$. But to actually form this set, we need something special, like a way to legally replace each $n$ in the ordinal $\omega$ (a set that exists by the axiom of infinity) with the corresponding surreal. This almost certainly requires replacement (and if it doesn't for some technical reason, I'm certain it'd require it by the time you get to $\omega+\omega$ for set theory reasons).
