What exactly did they do?

Bian and Booker produced an approximation to a generic (meaning non-self-dual) Maass form on SL(3,Z). Specifically, they computed an approximation to the eigenvalues and approximations to the first few hundred Fourier coefficients. It is believed that these coefficients are transcendental numbers. Previously it was known how to directly construct transcendental forms by transferring examples from lower rank groups, but this is the first indirect construction.

In their initial run they found 4 generic Maass forms, and one self-dual form. The self-dual form corresponds to the smallest symmetric square lift from SL(2,Z).

Their method is based on the GL(3) converse theorem, so one has high confidence in the answer. Additional evidence is that the L-function satisfies the Riemann Hypothesis for the first several zeros.

The calculation required solving many (non-sparse!) system with 10,000 unknowns, which is now within the capabilities of a personal computer. One of the steps was to figure out how to form such a large system that is sufficiently well-conditioned that you don't lose all digits of accuracy when you solve it. Their calculations are done in double precision, and the have maybe 6 decimal place accuracy in the answer. (Actually, a few more digits in the eigenvalues, less in some of the coefficients).