I am philosophically, and paradoxically amused by the concept of striving to enter rest and still, I get it. I get it on the day before every day off or vacation. I get it every day when I try to get my ducks in a row so that I can go home and not think about work (which does not always "work."). I get it; we concentrate, focus, and strive for rest because rest is not defined by the absence of activity as much as the enjoyment of fulfillment when everything for which we have be striving comes together.
The key is subtle. We do not achieve our rest. We enter it.
"Let us therefore strive to enter that rest, so that no one may fall by the same sort of disobedience. For the word of God is living and active, sharper than any two-edged sword, piercing to the division of soul and of spirit, of joints and of marrow, and discerning the thoughts and intentions of the heart. And no creature is hidden from his sight, but all are naked and exposed to the eyes of him to whom we must give account." - (Hebrews 4:11-13 ESV)