Allusion adds to the meaning of the third and fourth stanzas of “A Poison Tree” by putting the speaker in the position of Satan in the Garden of Eden.
Just as Satan gained enormous satisfaction from getting Adam and Eve to eat of the Tree of Knowledge, thus introducing sin and death into the world, so, too, is the speaker incredibly glad to see his enemy lying dead beneath the tree, having eaten the poisoned fruit.
This biblical allusion highlights one of the main themes of the poem: that anger and hatred can cause considerable harm. Satan's corruption of Adam and Eve was borne out of hatred for God's creation and anger at God himself.
Knowing that he would never be able to challenge God head-on, Satan set out to pollute the divine creation from within by getting Adam and Eve to defy the Almighty by eating the forbidden fruit.
The speaker has corrupted his own soul by bottling up his anger and hatred. His soul, graciously given to him by God, has been warped and polluted, just as the Garden of Eden was so despoiled by Satan disguised as a serpent.