Comprehension question regarding transcendent elements and the rational function field

Question

I'm currently reading through some stuff related to transcendental (or non-algebraic) elements. There I encountered the example that for a field K and K(x) = Q(K[x]) (the rational function field in one variable over K) the variable x is transcendental over K. Moreover any element K(x)\K is transcendental. I think I understand why x is transcendental since obviously for any non-zero polynomial $ f \in K[x]$ f(x) is non zero for some value of x but I don't really understand in what sense any element of K(x)\K is transcendental. Such an element would look like $f+K=\{f+k:\forall k\in K\}$ with $ f \in K[x]$, right? If we take a polynomial $ h \in K[x]$ how do we evaluate $h(f+K)$ and show it to be non zero? Also, in what sense would K(x)\K be a field extension of K since that seems to be included in the definition of transcendental/algebraic elements? I feel like I'm either misunderstanding certain definitions or I'm missing some.