Shakespeare suggests in this, as in many of his other sonnets, that the poem makes the subject immortal because the poem itself is immortal. The sonnet is not the words printed on any piece of paper; it existed before it was ever written down. It had to be created in Shakespeare's mind before he wrote the words on paper and before those words were ever printed in a book. Even if all copies of the sonnet were destroyed, the words themselves would still exist and could easily be written down again by people who remembered them. As here, for example:

Not marble, nor the gilded monuments
Of princes shall outlive this powerful rhyme,
But you shall shine more bright in these contents
Than unswept stone besmear'd with sluttish time.

Shakespeare might even have felt that once the words had been arranged in that particular form they would continue to exist even if all copies were destroyed and everyone forgot them. This is probably why he considers his poem a "living record," as opposed to a stone memorial. He might even have thought that the words existed in that form even before he found them, just as Michelangelo felt that the statue he wanted to create already existed in the block of marble and all he had to do was chip away the excess pieces. 

Not marble nor the gilded monuments...shall outlive this powerful rhyme...

Shakespeare wrote 154 sonnets which covered many subjects.  The first half of the poems were dedicated to a young man that Shakespeare loved. "Sonnet 55" finds its place in these poems and among the great works in literature.

Many of Shakespeare’s sonnets spoke to the idea of immortality. This poem’s entire purpose is to exalt the subject.  Not a lot of information has been found to substantiate exactly who the young man is that Shakespeare admires so greatly, but so many of his sonnets denote his grandeur.

In essence, this sonnet states with confidence that no marble or gold statue will outlive this commanding verse. The poet will immortalize the person, an unknown handsome lord who is not described in this poem. This young man [But you shall shine more bright] will last longer in this poem than the stone statues and monuments, which eventually will fade and be covered with dust. When destructive war upends the statues and the buildings, neither the god of war nor his fires will be able to destroy this testimony dedicated to you.

Nor Mars his sword nor war’s quick fire shall burn

The living record of your memory. 

The living record is the verse or sonnet that is written to immortalize the young man. Shakespeare believed that as long as the poem was read by others the man would live forever.

Death and all its hostility will not stop him. All of the future generations even to the end of time will praise him.  Therefore, until Judgment Day when he will be taken to heaven, this poem will be a tribute to him and live forever in the lovers’ eyes who read this poem.

Shakespeare often wrote as though his  poetry had a life of its own, as he does in this verse. What a lofty tribute to a person when the poet says that the man is so wonderful that he will outshine great statues! As long as this poem is read by future generations, this person will continue to live.  Shakespeare might be surprised to know that his prediction came true: We are still reading this poem four hundred years later.