The saint's whose feast we celebrate today is not someone we hear much about in Scripture. Before Our Lord Ascended into heaven, He gave the Apostles a mission; they were to go and bear witness to Him and spread His teachings to all corners of the earth. The Apostles, however, were one short of the twelve they began with. There are I am sure many reasons why they felt they needed to fill Judas' place. Perhaps since Jesus chose twelve to begin with, they felt that they should keep that number. I also think that because they were of Jewish origin that they felt the connection to the twelve tribes of Israel. Perhaps, unbeknown-st to us, Our Lord even told them to choose a replacement for Judas. All of that is really not important for us to know.
The story of how Matthias was chosen can be found in today's Mass reading in Acts 1:15-17, 20-26. We know that the principle reason Matthias was chosen was that he was a witness to Jesus' Resurrection.
Let us ask this saintly Apostle's intercession in helping us to bear witness to Christ and His teaching in our world today.