Our gemara on amud beis tells us that a convert requires a court of three people to preside over the conversion in order for it to be valid.  Likuttei Halakhos (Yoreh Deah, Laws of Slaves, Chapter 2:12) offers a mystical interpretation for why conversion requires this process.  Below is my best effort to distill and translate his esoteric ideas:

The scriptural derivation is that the verse uses the word “Judgment” juxtaposed to mentioning a convert, as the gemara explains.  On a deeper level, we understand that the function of judgment and justice is to clarify and resolve that which is false from that which is true.  Each party has a claim, and it is up to the judges to sort out the truth. 

The process of clarification of truth from falsehood is deeply wired into the fabric of the universe. We may say that originally, before creation, God is by Himself enveloped in utter truth.  In order to create the world and have material existence, there is an automatic injection of falsehood.  Finally, as the created beings act upon their existence and volitionally move back toward truth, i.e., move toward God and spirituality, we have the ultimate resolution.  Thus, it is a three stage process, so to speak, right, left, and then back to the middle. 

This tripartite pattern is represented in the three patriarchs, the three castes in Judaism (Cohen, Levite and Israel), and also in the three judges of a court.  So too, the convert must move away from his falsehood and embrace the truth.  Therefore, this is an enactment of justice and requires a tribuneral.