Why do we say a longer Tachanun on Mondays and Thursdays compared to the other days of the week?
I found three reasons in Aruch HaShulchan 234. Here's my summary:
These are days of acceptance (רצון). The reason for this being that Moshe Rabeinu ascended Mt. Sinai for the second tablets on Thursday and descended on Monday.
These are days of judgment (דין). The courts in heaven and the earthly courts convene on these days. The sign for this is that the Maadim (Mars).
Three zekanim (scholars) who were miraculously saved from a bishop who tried to burn them established this minhag.