The "rest" spoken of in Hebrews chapter 4 is God's particular rest for his children who will enter into glory. Ch.2 vs.10 introduces the concept of the Son of God "bringing many sons unto glory". He delivers from the fear of death, and from bondage, so that those "who are partakers of the heavenly calling" (ch.3.vs.1) have the belief that enables entrance into God's "rest". All of chapter 3 shows that it was unbelief that prevented most Israelites from entering into God's rest, i.e. entering in to the promised land. This is used as a parallel for Christians to grasp how belief is key to entering into God's rest, which has moved on from living on an earthly parcel of land to being in glory - heaven.
Chapter 4 enlarges on this. You ask particularly about verses 9-11 which clearly show that the topic has moved on from anything earthly to that which is spiritual. Rest is given by Christ Jesus, to those who believe in his finished work of redemption, that it now applies to them who believe only in him (Luke 24:21). Thus they can "cease from their own works". Their own works can never earn them redemption, or glory, or peace. Christ redeems spiritual Israel, and they know it. That is how he gives peace, but not as the world gives peace (John 14:27 & 16:33). But if they rest by faith in the peace that Christ gives, they experience spiritual rest even now, for their hearts are no longer troubled while on Earth. They will experience peaceful rest supremely when they enter into heaven where all striving has ended, as stated here:
"Here is the patience of the saints: here are they that keep the commandments of God, and the faith of Jesus... Blessed are the dead which die in the Lord from henceforth. Yea, saith the Spirit, that they may rest from their labours; and their works do follow them." Revelation 14:12-13
The answer to how one enters into that rest is in verses 2 and 3:
"For unto us was the gospel preached, as well as unto them: but the word preached did not profit them, not being mixed with faith in them that heard [the gospel]. For we which have believed do enter into that rest." Hebrews 4:2-3 - emphasis mine
CONCLUDING EDIT - Just as unbelief prevents anyone entering into God's rest (vs. 11), so faith in the finished work of Christ at Golgotha ushers that believer in to rest from their own works, or efforts. They 'rest' in Christ. It could not be clearer as in Romans 3:27-28, where it is not by the law, or of works, "but by the law of faith" and that saving faith is a gift of God, therefore entering into his rest, is the gift of God - to be received with deep gratitude.