The complete record of the birth of Moses has been divided into two parts. Details covering the birth are placed first (Exodus 2:1-10); his genealogy is second (Exodus 6:14-25). Each of the two records serves as a type of introduction to the two primary phases of his life. After his birth Moses lived in Pharaoh’s house as the son of Pharaoh’s daughter. After encountering God on Mount Horeb Moses became the prophet sent to bring the Israelites out of Egypt. The two-fold record also parallels the main authority in Moses life. At first, Moses was subject to the decrees of Pharaoh. Later, Moses became subject to the voice of the LORD God of Israel.
When Exodus is taken as the work of a single Author the placement of the genealogy can be studied as a purposeful choice by that Author. By placing the genealogy immediately before Moses and Aaron bring the plagues to Pharaoh, the effect is to make the point that bringing the people out of Egypt was God’s purpose for Moses life:
“Before I formed you in the womb I knew you, and before you were born I consecrated you; I appointed you a prophet to the nations.” (Jeremiah 1:5 ESV)
What was true for Jeremiah was true for Moses. Jeremiah began his call as a youth; Moses began his when he was 80-years old. What the LORD states directly to Jeremiah, He communicates (by implication) when He first separates the details of the birth and then places the genealogy before the conflict with Pharaoh.
The genealogy could be placed at the beginning of his life or the end and accurately communicate ancestry. When it is placed before the contest between the LORD and Pharaoh, it has the effect of stating this is where Moses truly begins the life he is called by God to live. In a sense it is a way of showing Moses' rebirth.
The source theory is that Exodus 6 is from the priestly source and is a redundant and inconsistent variant of Exodus 3:1 to 6:1 (JPS Study Bible p.115). Yet there is a significant difference between the two records:
Exodus 3:16-17: Go and gather the elders of Israel together and say to them, ‘The LORD, the God of your fathers, the God of Abraham, of Isaac, and of Jacob, has appeared to me, saying, “I have observed you and what has been done to you in Egypt, and I promise that I will bring you up out of the affliction of Egypt… (ESV)
Exodus 6:6-7: Say therefore to the people of Israel, ‘I am the LORD, and I will bring you out from under the burdens of the Egyptians, and I will deliver you from slavery to them, and I will redeem you with an outstretched arm and with great acts of judgment. I will take you to be my people, and I will be your God, and you shall know that I am the LORD your God, who has brought you out from under the burdens of the Egyptians. (ESV)
The second contains the four promises remembered at the Passover and specifically that the LORD will be their God. What begins as "the God of your fathers, the God of Abraham…" becomes “I will be Your God.” So the prophetic message of Moses is not simply to bring the Israelites out of Egypt (Exodus 3); it is tell the people of the relationship where they will be His people and He will be their God (Exodus 6). Connecting the genealogy to the second message serves to reinforce the significance of this message.
There is additional evidence that the arrangement of the genealogy and birth details of Moses is a purposeful work of a single Author. The question is based on the fact that the information is not arranged as is expected. Logically the genealogy, names of the parents, and birth details should be together in a single account. While the information can be divided it should be kept together as an integral unit:

In Exodus the division is such that all of the names have been kept together and separated from the details of the birth. At first something is missing: the Author did not identify the parents or the ancestors. Seemingly important information has been left out. When those facts are later revealed, the reader is able to go back and reconstruct the complete record:

When this arrangement is taken as the work of a single Author, additional observations about His methodology can be made:
The Word of God has been rightly divided (one is Moses’ mother, the other his father).
When properly divided and combined, what was first must be placed last and what had been last is placed first.
The two records can be combined into one.
These elements are consistent with principles taken from Scripture:
Be diligent to present yourself approved to God, a worker who does not need to be ashamed, rightly dividing the word of truth. (2 Timothy 2:12 NKJV)
And indeed there are last who will be first, and there are first who will be last.” (Luke 13:30 NKJV)
While Scripture can be viewed narrowly as different works of human authors whose words were later assembled into the record we find today, it can also be seen as the work of a single Author applying His unchanging truth from beginning:
And the Word became flesh… (John 1:14 NKJV)
Therefore a man shall leave his father and mother and be joined to his wife, and they shall become one flesh. (Genesis 2:24 NKJV)
The two records of Moses birth have been purposely divided into two records, one from his mother’s perspective (female) and one from his father’s (male). If rightly divided, the two can be made into one. When that is done the first is last and the last is first.
Therefore, the placement of the genealogy in Exodus is not only purposeful to give emphasis to the most important phase of Moses life, the separation of the birth and genealogy was accomplished using principles taken from both the Old and New Testaments. It is the work of a single Author.