The database of Belyi maps was constructed by Michael Musty, Sam Schiavone, Jeroen Sijsling, and John Voight. A detailed description of its construction can be found in [arXiv:1805.07751].
The underlying algorithm (numerical computation of modular forms on triangle groups) that was used to compute Belyi maps can be found in
Numerical calculation of three-point branched covers of the projective line, by Michael Klug, Michael Musty, Sam Schiavone, and John Voight, LMS J. Comput. Math. 17 (2014), no. 1, 379-430, [MR:3356040].
If you use the database of Belyi maps in your research please cite:
A database of Belyi maps, by Michael Musty, Sam Schiavone, Jeroen Sijsling, and John Voight, Proceedings of the Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII), eds. Renate Scheidler and Jonathan Sorenson, Open Book Series 2, Mathematical Sciences Publishers, Berkeley, 2019, 375-392 [MR:3952023],
as well as any of the references cited above that are of particular relevance.