I have proved that the surreal numbers have the properties of a real closed field. Now I should be able to explain what the importance of this real closure is. unfortunately I do not have a background in model theory. I read somewhere it has to do with quantifier elimination, but can someone explain to me the importance of the real closure vs the algebraic closure in a non model-theoretic way?
The difference between real closed and algebraically closed fields isn't really rooted in model theory. Recall that a field $K$ is algebraically closed if every nontrivial polynomial over $K$ has a root in $K$. Note that if $K$ is actually an ordered field, $K$ can't be algebraically closed, since $x^2 = -1$ can't have a solution (exercise: the square of any number in an ordered field is non-negative). So, if we're working with ordered fields, the appropriate notion is to be real closed: every polynomial over $K$ that might have a solution in an ordered field has a root in $K$.
Phrased another way, an ordered field $K$ is real closed if every element in every extension of $K$ as an ordered field is transcendental over $K$.
The reason why we care about real closed fields is that these have very tame geometry. This does include quantifier elimination, which in this case tells us that a projection of a semialgebraic set is semialgebraic.
But there are many other nice properties stemming from the elementary equivalence. For example, you can apply more or less any theorem true about continuous real semialgebraic functions, e.g. they are differentiable almost everywhere and you have Taylor's formula.
Moreover, you can apply various methods specific to real algebraic geometry like Sturm's algorithm, real nullstellensatz etc.
Much of this is analogous to algebraic closure, except in that case geometry is even more tame (like that of the complex numbers), but topology is not definable, which makes it weaker in a sense. It does not make that much sense to talk about continuous functions over an arbitrary algebraically closed fields.
This all is in stark contrast to, say, the field of rational numbers which is provably very much not tame.