This has caused me to study more about this phrase and how it relates to verses in the New Testament. Here are a few ideas:
(1) Paul quotes Psalms 95 and writes quite a bit about entering God's rest in Hebrews 3 and 4, for example:
1Therefore, since the promise of entering his rest still stands, let us be careful that none of you be found to have fallen short of it. 2For we also have had the good news proclaimed to us, just as they did; but the message they heard was of no value to them, because they did not share the faith of those who obeyed. (Hebrews 4, NIV)(2) The Greek word translated into "rest" in Hebrews 3 and 4 is κατάπαυσις (katapausis). which is described as:
the heavenly blessedness in which God dwells, and of which he has promised to make persevering believers in Christ partakers after the toils and trials of life on earth are ended (Source)
(3) This adds context to the words of Jesus:
28 ¶ Come unto me, all ye that labour and are heavy laden, and I will give you rest.
29 Take my yoke upon you, and learn of me; for I am meek and lowly in heart: and ye shall find rest unto your souls.
30 For my yoke is easy, and my burden is light. (Matthew 11)