This sonnet, Milton's first published poem, says that no monument or "relic" built to honor Shakespeare can ever rival Shakespeare's own work. His plays and poems are his monument. His words, flowing easily from his pen, are more magnificent and enviable than the tomb of any king: in fact, even kings would wish to entombed (preserved) in the "marble" of Shakespeare's words.

In the first six lines of the sonnet, Milton questions what need there is to build a monument to Shakespeare. He finds the whole idea puzzling. Why erect such a "weak witness" to the writer's name?

In the last eight lines, Milton continues by stating the Shakespeare built his own monument through the body of work he produced. His words impress themselves on us deeply.

One imagines that Shakespeare, who wrote sonnets expressing the idea that he was immortalizing his loved ones in verse, would appreciate this sonnet.

The sonnet “On Shakespeare” is a tribute from Milton to Shakespeare. 

What needs my Shakespear for his honour'd Bones,
The labour of an age in piled Stones,
Or that his hallow'd reliques should be hid
Under a Star-ypointing Pyramid?

The first four lines are saying that Shakespeare does not need a pyramid ("Star-ypointing pyramid") that takes ages to build in order to honor his life.  Remember that a pyramid is a tomb that took a very long time ("the labour of an age") to build.

Dear son of memory, great heir of Fame, 
What need'st thou such weak witnes of thy name?
Thou in our wonder and astonishment
Hast built thy self a live-long Monument.

The next four lines state that building Shakespeare a monument would be pointless.  He does not need a monument (“what need’st thou such weak witness of thy name?”) because he has already built the monument for himself.

What is the monument he has built?  The next four lines are an important clue:

For whilst to th' shame of slow-endeavouring art,
Thy easie numbers flow, and that each heart 
Hath from the leaves of thy unvalu'd Book,
Those Delphick lines with deep impression took,

Milton says that while other poets are slow (“the shame of slow-endeavouring art”), Shakespeare’s poetry flows easily (“thy easie numbers flow”).  He also goes on to say that the reader takes from what Shakespeare has written a meaning that may come from the gods. (Delphick is an allusion to Delphi,  where Apollo, the Greek God of poetry, visited the oracle).  His monument, therefore, is his work.

The poem ends by telling us that Shakespeare is in a tomb a king would envy:

Then thou our fancy of it self bereaving,
Dost make us Marble with too much conceaving;
And so Sepulcher'd in such pomp dost lie, 
That Kings for such a Tomb would wish to die.

The reader him or herself is the tomb. When we read Shakespeare’s words (“our fancy of it self bereaving”) we are amazed (“dost make us Marble”); we become the material of his tomb (marble).  He does not need a pyramid because his words produce everlasting memory.  Kings may have pyramids, but they don’t have that.