Shakespeare is essentially saying that the sonnet he is writing to someone he loves will live longer than any existing monuments. He was exactly right, because this sonnet, as well as his others, have survived for some four hundred years and will very likely survive for centuries longer.
There are two interesting features of this particular sonnet. One is that it appears to be addressed to someone who has recently died and is therefore an elegy. This would explain why the poem opens with graveyard imagery, and would also explain the concluding couplet: "So, till the judgment [i.e. Judgment Day] that yourself arise, / You live in this, and dwell in lovers' eyes." If the person being addressed were not dead, the sonnet would seem unusually morbid and inappropriate.
The other interesting feature is the remarkable image contained in the lines: "But you shall shine more bright in these contents / Than unswept stone, besmear'd with sluttish time." He is enclosing a metaphor within a simile. Old neglected marble headstones do resemble marble front steps of houses which are not properly cleaned. The steps should be swept to remove dirt and debris and then scrubbed, but a sluttish maid will simply mop them and leave a brownish-greenish film on them similar to that seen especially on headstones that are flush with the earth.
The reference link below offers further explication of Sonnet 55. I also recommend that you compare this sonnet with Sonnet 19.