Hebrews 6:13-18 meaning

The Pauline Author takes the time in this passage to show the supreme sovereignty and honesty of God. He has made many promises, and He does not break His word. The goal here is to motivate the faith of the Hebrew readers, that they can put their trust in the most trustworthy of all beings — God. By understanding God's sincerity, we can have assurance that our sufferings for Christ's sake will be worth it in the end, and we will be rewarded for our faithfulness.

The Pauline Author is specifically referencing God's promise to Abraham—For when God made the promise to Abraham, since He could swear by no one greater, He swore by Himself (v 13). He references the verses found in Genesis 22:16, 18: "By Myself I have sworn, declares the Lord, because you have done this thing and have not withheld your son, your only son...In your seed all the nations of the earth shall be blessed, because you have obeyed My voice." 

The Pauline author explains the phrase swore by Himself: For men swear by one greater than themselves, and with them an oath given as confirmation is an end of every dispute. In the same way God, desiring even more to show to the heirs of the promise the unchangeableness of His purpose, interposed with an oath (vv 16-17).

The context of this oath from God takes place after Abraham was commanded to take his son, Isaac, up on a mountain and sacrifice him to God. Abraham obeyed, believing God would raise Isaac from the dead (Hebrews 11:17-19).

But before he could take his son's life, God interceded, telling him to spare Isaac, and provided a ram to be offered as a sacrifice instead. God then swore by Himself saying "I will surely bless you and I will surely multiply you (v 14)." He promised Abraham that He will bless him and  give him as many descendants as there are stars, and that all the nations of the world will be blessed by the nation of his descendants. This oath to bless Abraham is made by God "because you have obeyed My voice" (Genesis 22:18). And so, having patiently waited, he [Abraham] obtained the promise (v 15). 

The Pauline Author uses Abraham's example of obedience to encourage his readers that God does not lie. If we take hold of the hope set before us, we are heirs of the promise and partakers in the unchangeableness of His purpose. The result of choosing this perspective is that by two unchangeable things in which it is impossible for God to lie, we who have taken refuge in the promised blessings of God would have strong encouragement to take hold of the hope set before us (v 18).

The promised blessings for those who wait patiently and endure as Abraham endured include reigning with Christ in His coming kingdom. Romans 8:17 talks about being "fellow heirs with Christ, if indeed we suffer with Him so that we may also be glorified with Him." Christ's inheritance is that He will rule all creation, as it says in Philippians 2:9-10: "God highly exalted Him, and bestowed on Him the name which is above every name, so that at the name of Jesus every knee will bow." 

This name is Son, which was stated at the beginning of the book of Hebrews: 

"having become as much better than the angels, as He has inherited a more excellent name than they. For to which of the angels did He ever say, 'You are My Son, today I have begotten You'?"
(Hebrews 1:4-5)

We too can become sons who are rewarded to share in the Son's glory, if we follow His example and do not fear the world's rejection, instead living out a life of continual sanctification (a life apart from the world's ways, in trusting obedience to God's good design, just as Christ lived), then the Son will boast of us in His Father's presence, and we will share in Jesus's glory (Hebrews 2:10-12).

God's plans do not change. He does not break His promises. We can rest assured He will bless us if we are patient and obedient to His voice, just like Abraham.