There are several universality theorems for random matrices. The theorem that best captures what universality means is sine kernel universality "in the bulk" for unitary ensembles.
This theorem basically tells you about conditional eigenvalue densities. So, let's first look at an EXACT conditional density for eigenvalues of a 10 by 10 GUE matrix.
The red curve is the mean density of eigenvalues.The blue curve is the conditional mean density given that some other eigenvalue is 1.
Explanation: the eigenvalues behave line charged particles confined in a potential, so the eigenvalue at 1 repels other eigenvalues. This is an example of "level repulsion".