These lines actually describe the way in which humans live their lives and how the death of someone important impacts them. As if to reassure the reader, the speaker in the poem talks of the way that the reader's death will not change anything in the large scheme of things. Note what it says:
the gay will laugh
When thou art gone, the solemn brood of care
Plod on, and each one as before will chase
His favourite phantom...
Mark the way that "chase his favourive phantom" is a metaphor to describe the frivolous way in which we live our lives and how we pursue illusory or ephemeral objects. Life itself is depicted as "the solemn brood of care" which "plods on," making life feel as if it is nothing more that a massive toil to be endured. Such descriptions therefore support the central theme of the poem: that death is something to be looked forward to and something to be embraced, as it gives us rest from such "plodding," rather than feared.