In the last two lines of Shakespeare's Sonnet 18, the speaker argues that his beloved will be immortalized by the poem, that they will live on in the minds of men long after they have died.

This explains why it is inappropriate to compare the speaker's beloved to a summer's day or all the other features of the natural world referred to elsewhere in the poem. A summer's day, no matter how beautiful, will eventually fade away. And summer itself, no matter how hot and sunny, will always pass on to autumn.

But the beloved's “eternal summer,” their inner beauty as a human being, will never fade away, will never die. That is because the speaker has immortalized them in verse, in “eternal lines” that will live on so long as they are read and enjoyed by successive generations of poetry lovers.

In effect, this constitutes a great victory of art over life: Ars longa, vita brevis; art is long, life is short. In other words, art, like Sonnet 18, transcends the time and place in which it was written to speak to successive generations. Therein lies its universality, what makes it truly special and enduring. The same cannot be said for a summer's day, no matter how beautiful.

The last two lines of Sonnet 18 are as follows:

So long as men can breathe or eyes can see,

So long lives this, and this gives life to thee.

A Shakespearean sonnet usually ends with a "turn" in the final couplet. This means that the subject of the first twelve lines is considered from a different perspective, or that the poet adds a new idea to the argument he has been making. In Sonnet 18, the poet has been comparing his beloved with a summer's day, to the advantage of the beloved. None of the factors which might spoil the summer's day are present in the addressee, who outshines it in every way.

The final point Shakespeare makes is that unlike the summer's day, the subject of his sonnet is immortal. This is a larger claim than the reader expects. It is true that young lovers last longer than summer's days, but they do not last forever. The poet, therefore, explains in the last two lines,

So long as men can breathe or eyes can see,

So long lives this, and this gives life to thee.

What the last two lines of this sonnet mean is that Shakespeare is bragging about the importance of his work and of this poem in particular.

In the rest of the poem, he has talked about (among other things) how brief and transient a summer's day is.  Then he has contrasted that with how his love will be immortal.  He has said that she will never die because he has written this poem about her (that is what the line just before the couplet is saying).

In the couplet, he completes the thought by saying that as long as people exist, this poem will exist and she will live in the poem.