What does it mean that we, the regenerate, are Christ's Bride?

Christ is full of grace, life, and salvation. The soul is full of sins, death, and damnation. Now let faith come between them and sins, death, and damnation will be Christ’s, while grace, life, and salvation will be the soul’s; for if Christ is a bridegroom, He must take upon Himself the things which are His bride’s and bestow upon her the things that are His. If He gives her His body and very self, how shall He not give her all that is His? And if He takes the body of the bride, how shall He not take all that is hers?..

By the wedding ring of faith He shares in the sins, death, and pains of hell which are the bride’s.  As a matter of fact, He makes them His own and acts as if they were His own and as if He Himself had sinned; He suffered, died, and descended into hell that He might overcome them all.  
Now since it was such a One who did all this, and death and hell could not swallow Him up, these were necessarily swallowed up by Him in a mighty duel; for His righteousness is greater than the sins of all men, His life stronger than death, His salvation more invincible than hell.  
Thus the believing soul by means of the pledge of its faith is free in Christ, its bridegroom, free from all sins, secure against death and hell, and is endowed with the eternal righteousness, life, and salvation of Christ its bridegroom.  So He takes to Himself a glorious bride, ‘without spot or wrinkle, cleansing her by the washing of water with the Word’ [cf. Eph. 5:26-27] of life, that is, by faith in the Word of life, righteousness, and salvation.  In this way He marries her in faith, steadfast love, and in mercies, righteousness, and justice, as Hosea 2:19-20] says:

'And I will betroth you to Me forever. I will betroth you to Me in righteousness and in justice, in steadfast love and in mercy. I will betroth you to Me in faithfulness. And you shall know the Lord.'

Her sins cannot now destroy her, since they are laid upon Christ and swallowed up by Him.  And she has that righteousness in Christ, her husband, of which she may boast as of her own and which she can confidently display alongside her sins in the face of death and hell and say, ‘If I have sinned, yet my Christ, in whom I believe, has not sinned, and all His is Mine and all mine is His,’ as the bride in the Song of Solomon 2:16 says,  

‘My beloved is mine and I am His.’

~ Martin Luther