in 1678, published in 1682) was written to counter several attacks on Dryden's poetic and dramatic skills, as well as his politics, by Thomas Shadwell, a playwright who also happened to be Dryden's political opposite.  Several of Shadwell's attacks were personally abusive, and Dryden's ultimate response to Shadwell, whom Dryden believed was not only his social inferior but a writer without true talent, was Mac Flecknoe.

Because the poem is a mock-epic, its language and subject must be lofty and mimic the conventions of epic poetry, but its subject is essentially trivial.  The loftier the diction and treatment of the subject--in this case, Shadwell--the greater the satiric effect.  The two lines you quoted above are meant to establish the tone of an epic with a seemingly profound statement about the brevity of human life, but these lines only serve to set up the framework of Dryden's satire:

All humane things are subject to decay,
And, when Fate summons, Monarchs must obey:
This Fleckno found, who, like Augustus, young
Was call'd to Empire, and had govern'd long. . . . (ll. 1-4)

Richard Flecknoe was a poet and writer of travel literature, but his verses, which are not terrible, evoked some laughter among other poets, such as Andrew Marvell, and later, Dryden, when he used Flecknoe in his mock-epic as the poetic father of Thomas Shadwell.  The opening four lines describe the death of Flecknoe and compare him to one of the greatest Roman emperors, Augustus, a clear sign that what follows will not be flattering to Shadwell, who is described a few lines later as 

who most resembles me:
Sh-- alone my perfect image bears, 
Mature in dullness from his tender years. (ll. 15-16)

The title Mac Flecknoe is meant as a kind of parody on the Gaelic naming convention for a son--Mac--and so the title is The Son of Flecknoe, and Shadwell is the heir of a writer of bad (in Dryden's view) verse.

The opening lines of the poem, then, are meant to establish the tone of an epic, and that is their only purpose.  The meaning of the lines, of course, can be taken as profound, but that is Dryden's way of setting Shadwell up for a poetic fall.

In John Dryden's satirical poem Mac Flecknoe, the following lines can be found:

All human things are subject to decay,
And, when Fate summons, monarchs must obey.

The quote refers to the fact that no one, not even monarchs, can stop death when it comes.

The lines can be meant to detail the fact that all of mankind will, at one time or another, succumb to death ("subject to decay"). The reference to fate (something unavoidable) details the fact that death is, naturally, unavoidable.

Dryden also mentions monarchs in the lines. This reference is important given that they (monarchs) were seen as being the most powerful at the time of the text's writing. That being said, even monarchs did not have the power to stop death.

Outside of the human aspect of death, Dryden is also referring to not only humans, but all "human things." Human things refer to those items which were created by humans (which means both life, through birth, and all man-made objects). Therefore, Dryden is basically saying that all things on earth, created by man, will fall to decay. The only things which will not fall to decay are those things not created by man.