Rest for the weary who toil in the desert of their day to day lives is a significant comfort. In chapter 4 verses 1 through 13 the author continues this discussion of rest. Remember, his argument to this congregation who is experiencing some who have reverted back into their former ways and others who are considering it is that Jesus is more superior than Moses. The new covenant is more superior than the old because it is in the new covenant that we find actualized which was formerly in the old covenant type and shadow. So his argument continues that the Sabbath rest is also actualized in the new covenant.
In 4.1 having just quoted Psalm 95 in reference to Exodus 17 where Israel rebels against God and is not permitted to enter His rest or the promise land the author of Hebrews wants to ensure his readers understand that this promise to enter His rest still remains only as we are going to see it isn't a physical geographical location but rather it is a future and eternal rest. Moreover the author at this point introduces the idea of promise which becomes a constant theme through out the book of Hebrews. This is because it is God who issues the promise and is the primary basis of our faith. The problem in this congregation is there may have been some who don't care about the promises of God and some who question whether God will make good on His promise. What ever the case this is the promise of the new covenant a covenant of grace. That is the reason for the "fear" for there were some in the congregation that were acting like their rebellious ancestors as they rebelled against God in Maribah (rf. Exod.17 Ps.95). Why bring this up? The author of Hebrews is simply asking his readers to weigh the consequences of their disobedience, "do you truly want the Sabbath rest in this life and the life to come?