Consonance elevates the poem's language, giving it some meaningful music.
For instance, listen to the gentle repeating sounds in the first stanza:
Happy the man, whose wish and care
A few paternal acres bound,
Content to breathe his native air,
In his own ground.
Here, a subtle /r/ connects those "paternal acres," creating a fitting feeling of stability and continuity. And the unobtrusive /n/ and tapping /t/ consonance that threads these lines feels quiet and humble as this "happy" man's life.
Sibilance does similar work in the third stanza:
Blest, who can unconcernedly find
Hours, days, and years slide soft away,
All those gentle /s/ sounds feel whispery, peaceful, and smooth as the time that "slide[s] soft" away.
But perhaps the most notable passage of consonance (and alliteration!) appears in the final stanza:
Thus let me live, unseen, unknown;
Thus unlamented let me die;
Steal from the world, and not a stone
Tell where I lie.
Here, restrained /l/ sounds suggest the kind of quiet life the speaker dreams of living, as well as the "unlamented" death he imagines for himself. And tip-of-the-tongue /t/ sounds evoke an image of the speaker tip-toeing from this world, trying not to be noticed on his way out.