In "Thanatopsis" the speaker is addressing anyone who loves Nature, and he tells them to "Go forth under the open sky, and list / to Nature's teachings" (ll. 14-15) when they begin to feel the sad thoughts of death coming upon them.  By looking at all that Nature has to offer, there "Comes a still voice" (l. 17) that will show them what happens to a body after it dies.  First, Bryant is very logical and scientific about the dead--the Earth "shall claim / They growth, to be resolved to earth again...[and] shalt thou go / To mix forever with the elements" (ll. 22-26).  These lines are similar to the Biblical quote: "for you are dust and to dust you shall return (Genesis 3:19).  But Bryant goes beyond the simple explanation that one's physical body will return to Nature; he also explains that when a person dies:

                                     Thou shalt lie down
With patriarchs of the infant world,--the kings,
The powerful of the earth,--the wise, the good,
Fair forms, and hoary seers of ages past (ll. 33-36)

So when one dies, he will join all the greatest people that history has known because the Earth is "one mighty sepulchre" (l. 37), or tomb, that holds everyone that has come before us.  These are the sentiments that the speaker offers to anyone who is thinking gloomily about death.