I didn't include the whole chapter of Genesis 38 but feel free to read it here.
The story of Joseph is interrupted in chapter 38 with the short story of Judah and Tamar, then it returns to the story of Joseph in chapter 39. Here the interrupting story does a couple of things to move the narrative forward.
First, it shows how Judah's life kind of spirals out of control. Remember he is the one who suggested to his brothers that they sell Joseph into slavery instead of kill him, not out of compassion or a desire to spare his life but so they would profit off of him. In the events of chapter 38 we see many of Judah's moral failings, including but not limited to: 1. sleeping with a prostitute, or so he thought, it was actually his daughter-in-law 2. failing to have his youngest son fulfill his levirate marriage duties, and 3. seeking to have his daughter-in-law punished for infidelity when he was the one who committed the act with her. Towards the end of the story Judah confesses to everyone present, "[Tamar] is more righteous than I..." (38:26)
Second, this story continues the pattern of the younger ruling the older. The narrative of Genesis has already set this pattern with Isaac and Ishmael and Jacob and Esau. In spite of this history in the family, Joseph's older brothers were still furious over the thought of Joseph ruling over them. His dreams of his brothers bowing down to him were the final push that led them to take action against Joseph. In an ironic (ie. divine action) twist at the end of the story, Judah's twins through Tamar confuse the birth order as well. Zerah puts his hand out first to which the midwife ties a scarlet thread indicating he was the first born. Then Perez actually came out first. Judah's line ends up carrying on through Perez (the younger) not Zerah (the older) (Matt. 1:3). So in this twist of events the very thing that he abused and sold Joseph for is happening in Judah's own family. The one whom God has chosen will rule, regardless of birth order, and there is nothing anyone can do to stop that plan from coming about.
Furthermore, at the end of his life when Jacob delivers his blessings to his sons he names Judah himself as ruler of the family, passing up his 3 older brothers because of their failings and shame they brought to the family (Gen. 49:8-12). Eventually king David comes through the line of Judah and ultimately Jesus comes through the line of Judah.