Striving and resting seem to be opposites. One implies effort. The other implies some kind of ceasing of effort. The rest mentioned, we know from previous studies, is trusting in divine grace as it has finally been revealed in the Christ, God the Son, and His finished atoning work on the cross. It's a ceasing of merit-based righteousness for securing relationship with God. All references to the Old Covenant Sabbath are a picture of this great resting in the finished work of redemption just as the original Sabbath rest was a rest after the finished work of original creation.
Our text, perhaps to the surprise of many, indicates resting in Christ's gracious atoning work isn't the same as relaxing in Christ's finished work. No. There is a striving demanded in that eleventh verse - "Let us therefore strive to enter that rest..."
There is nothing in the exhortation of this text to indicate this striving is somehow contrary to divine grace. There is no hint of legalism here - no resorting back into some kind of works theology. No. Nothing of the kind. Here we have a divinely inspired, new covenant endorsed, Holy Spirit infused call to strive to enter into the rest of grace. We have to do something with this. It is, after all, a divine call to new covenant striving.