This is a version of the formal character.
It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ then it integrates to a finite-dimensional representation of $SL_2(\mathbb{C})$, and for any such representation we can compute its character in the group-theoretic sense. If the representation is given by $\rho : SL_2(\mathbb{C}) \to GL(V)$ then its character is
$$\chi_V(g) = \text{tr}_V(\rho(g)).$$
This is a continuous class function on $SL_2(\mathbb{C})$. Since diagonalizable matrices are dense it is determined by its behavior on diagonalizable matrices and hence on diagonal matrices (the maximal torus). Every such matrix has the form $\begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix}$ for some $x \in \mathbb{C}^{\times}$, so in fact the character is determined by its restriction to matrices of this form, where it becomes some function of $x$, which by an abuse of notation I will denote $\chi_V(x)$. For example, the character of the defining representation is
$$\chi_{\mathbb{C}^2}(x) = \text{tr} \begin{bmatrix} x & 0 \\ 0 & x^{-1} \end{bmatrix} = x + x^{-1}.$$
More generally you can compute that the character of $S^n(\mathbb{C}^2)$ in this sense is $x^n + x^{n-2} + \dots + x^{-n+2} + x^{-n}$.
This is pretty much the formal character already, except that $x$ is taken to be a formal variable rather than an arbitrary element of $\mathbb{C}^{\times}$. The reason for the funny notation $x^h$ is to suggest the intuition that we are invoking a formal version of the exponential map: namely, we could also have written an arbitrary diagonal matrix in $SL_2(\mathbb{C})$ as an exponential
$$\begin{bmatrix} e^z & 0 \\ 0 & e^{-z} \end{bmatrix} = \exp \left( z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right) = e^{zh}, z \in \mathbb{C}$$
of an element $zh$ of the Cartan subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ given by diagonal matrices. In relation to our previous parameterization we have $x = e^z$, so you can think of "$x^h$" as shorthand for the exponential $e^{zh}$. A more abstract interpretation of this notation where we really try to make $x$ a formal variable is also possible, I think. Basically $x^h$ should be the linear map with the property that if $v$ is an eigenvector of $h$ with eigenvalue $\lambda$ then $x^h v = x^{\lambda} v$.
