Heaven and earth were created by God, and God has the power to end them. In this regard, Jesus' proclamation that His words are more unchanging, solid, trustworthy, and permanent than the heaven and earth constitutes one of His strongest attestations to be an eternal being. By this statement, Jesus is reassuring His followers that His words are absolutely reliable and cannot be swayed in any way. 
As the scriptures tell us, heaven and earth will eventually pass away. By the Word of the Lord, the present heaven and earth will be reserved for fire, to be kept for the day of judgment, and to destroy all ungodly men. God is building a new heaven and a new earth, which will serve as the place for the rest of righteousness since the first heaven and the first earth will pass away, as He foretold in His Word.
The heaven and the earth will eventually pass away since this world has been contaminated by sin. The pure words of the Lord Jesus, however, are perfect. To fallen men, they are life and light and will never, ever be extinguished. God's Word endures forever, as well as the Words of His Christ for all eternity.

Dear Heavenly Father, thank You for sending Jesus, the risen, ascended, and glorified Word of God, whose words will never fail. Thank You for entrusting Jesus as the living Word of God, whose promises will endure for all time. I pray that I may hold fast to Your Word of truth no matter what happens in this world and no matter what trials and tribulations I may experience. As you have instructed, I will take the helmet of salvation and the sword of the Spirit, which is Your Word. 
In Jesus's name I pray. Amen.