Hebrews 3:6 stands at the end of a paragraph in which we are asked to "consider the Apostle and High Priest of our confession, Christ Jesus" (Hebrews 3:1). In the intervening four verses, the author of Hebrews, probably ultimately the apostle Paul, makes a comparison between Christ and Moses in terms of their faithfulness. Jesus is, of course, superior to Moses in many ways, but in the area of faithfulness, He is far greater because He is no mere servant, as Moses was, but the Son and Heir of His own house, the house of God.
A second distinction that the author makes is that, while Moses functioned as a faithful servant or steward of the house, Christ built the house. In other words, while Moses dutifully followed orders concerning the running of the house during his time of service, Christ gets all the credit for planning, designing, building, and maintaining the house, as He is its Creator. The author makes this plain in verse 4: "He who built all things is God."
So the author makes two major points: 1) Jesus Christ is the faithful Son of God and Heir of all things, and 2) He Himself is the Creator God, the One who made everything (John 1:3; Colossians 1:16). For these reasons, He is worthy of all glory and honor.
In verse 6, the object of our comments, the author brings Christians, the church, into the argument. We are the house of God that Jesus has been building and that Moses faithfully served. The Son of God has been faithfully working on us both individually and collectively since the beginning to fit us into His house—whether we wish to look at it as a building or a family—in the place that most suits us and where we will function the best for His purpose.
The emphasis here needs to be on the fact that He, appointed by the Father to this task, has executed His responsibilities faithfully in every respect. He never shirks a job, never does shoddy work, and never fails to finish what He starts. Jesus Christ always does perfect work.
So, as the verse implies, we should have perfect confidence and joy in our Creator in bringing us to salvation and eternal life. We have no reason to doubt! Our responsibility, then, is to "hold fast," to stand firm, to endure to the end, through whatever assails us in the meantime.
There is nothing that can stop Christ from finishing His work perfectly—except us. We can fail Him (see Hebrews 6:4-8; 10:26-31); we can prove unfaithful, which is why the author's next section is an exhortation to be faithful and a warning not to follow the unfaithful, unbelieving example of the Israelites in the wilderness.
To this end, he repeats his encouraging remarks in Hebrews 3:14, "For we have become partakers of Christ if we hold the beginning of our confidence steadfast to the end." We have to keep hanging on, faithful and trusting that God, in His perfect work, has everything under control. So Jesus Himself tells us in Matthew 24:13, "But he who endures to the end shall be saved."