Sorry to give a reference to my own paper, but perhaps what you are looking for is contained in section 2 of this paper; see also Theorem 1.3. The basic idea is that the Selberg and Kuznetsov trace formulae both involve spectral sums but with different weights. To get Selberg weights from Kuznetsov one needs to multiply by $L(1, \text{sym}^2 u_j)$ where $u_j$ is a Hecke-Maass form associated to the spectral parameter $t_j$. Thus one obtains a sum of Kloosterman sums of the form $\sum_{c} \sum_{n} \frac{S(n^2, 1;c)}{ c n}$ with some weight function. An application of Poisson summation in $n$ leads to values of quadratic Dirichlet $L$-functions at $1$ which by the class number formula can be expressed in terms of class numbers. These are then connected to the geometric side of the Selberg trace formula using Sarnak's thesis. In the paper referenced above this is carried out in the reverse direction.

In my personal experience, the Kuznetsov formula often gives stronger error terms with minimal extra work. The representation in terms of Kloosterman sums allows for algebraic geometry (the Weil bound) to enter the game as well as other ideas in the theory of exponential sums. The Kuznetsov formula is also very useful in the reverse direction for studying Kloosterman sums which occur naturally in many number-theoretic applications. One can examine the papers of Deshouillers-Iwaniec for many interesting examples.