First reason is that we can’t survive well in this world if we don’t call ourselves as exiles and aliens, because everything the world stands for is against us as Christians. Though there are common grace that have been shown in humanity including non-believers, it will be impossible to truly live as a victorious Christians without such mindset. First, we are a chosen race. We are not just any race, by race, bible means a spiritual race, not the physical race. We are a chosen race, chosen by the Almighty God even before the foundations of the world. Then we are the royal priesthood, royal means the King of kings, and we are priests to him (God) in serving and following him. I cannot see we as Godly priests without being exile and alien in this world. Then we are a holy nation, that is the deal breaker in many ways. Holiness is the last thing the world wants. Liberal progressive Christians hate holiness, they love the kindness, social justice, and compassion of Christ, but they hate the holiness of Christ. Why? Because holiness will restrict and restrain them form following their lust, and their free-lancing lifestyle and yet calling themselves as Christians, as a label for their real agenda, desecrating the holiness of God. The judgment of which coming will be severe. 

Then we are a people for God’s own possession, we are Christ’s. We belong to Christ, and he owns us. Ownership is a strong word, and the world hates it. They want to belong to themselves, their own inner self liberty, be themselves. The culture of the world will not allow such ownership, and hence we are to be aliens, and exiles. (Hebrews 2:9-10) 

And because of the above, as explained, its impossible to live the holy life, the royal life, the chosen life, God owned life, Hebrews exhorted us to be aliens and exiles to abstain from the passions of the flesh which war against our souls. (v11). The sense of belonging is a very strong factor here, to be indulged or involved in the ways of the world. The ways of the world, wages war against our souls, a constant antithesis to everything the Bible teaches. The social media, Hollywood movies, even books that our kids or ourselves study in colleges all point at a different value system than the Word of God. How do you maintain your sexual holiness when everything they talk about is pretty much looseness, licentiousness? And you go against the culture but not indulgencing with them (1 Peter 4:4). The only way is to be an alien, we don’t belong to them. And the other one is being an exile; we don’t belong to their citizenship either. We are exiles form heaven, only here temporarily. A good mindset to have as overtime we face temptations or hardship in this world, we know that our time here is only temporary, we are pilgrims, passing by, our real home and destiny is heaven!  

On top of that the expressions of our exile and alien in this world will be to submit to authority in this world (2:13), respect our masters or employers, bosses (v18), wives submit to husbands, and husbands honor your wives (3:1-7), and finally be prepared to suffer for righteousness sake (3:12-4:19) 

My own context will be having a mindset of being different from the world pattern, not for the sake of being different, but rather for the Lord’s sake who suffered for me, and hence I am called to suffer for him if and when necessary. Tio pay the price of being a Christian living with a different mindset and values we pursue in this world, like John Piper said don’t go for retirement seashells, but go for the calling of God in this world. The only thing that will truly satisfy me (us), is to know God’s will and calling and do it with all our hearts and see that fulfilled in this world where we are merely passing by, building for an eternity. And we will rejoice as we share Christ’s sufferings now, when his glory is revealed (v13).