An echo is a pretty lonely sound, as sounds go. Think about it: the only way to hear an echo is if you're in a big empty space (with some sort of distant walls or ceiling, of course). In this poem, though, an echo is a sign of connection, not isolation.
Sure, our speaker may be in a big empty space in the middle of a field, but he realizes that he's not truly along there—or anywhere for that matter. The sounds of this poem echo in various ways as subtle reminders of the connection our speaker has with the absent mower. Ready for some examples? Some examples? Some examples (did you get the echo there?):
But as I said it, swift there passed me by
On noiseless wing a 'wildered butterfly, (11-12)
In these lines, the speaker's no longer really alone in the field. He's now got a butterfly buddy. True to form, we see sound echoes here in the alliteration of S words like "said" and "swift," and W words like "wing" and "'wildered." Added to the mix is some consonance, with the repetition of the S sounds in "as," "passed," and "noiseless."
Pretty soon, we're getting sound echoes all over the place. We have T word alliteration: "And would have turned to toss the grass to dry;" (20). We have L word alliteration: "But he turned first, and led my eye to look" (21). And we have still more T word alliteration: "At a tall tuft of flowers beside a brook" (22).
In fact, as the poem develops, you can see alliteration and-or consonance in nearly every line, right up until the W word alliteration in the poem's final couplet:
'Men work together,' I told him from the heart,
'Whether they work together or apart.' (41-42)
"Work," "Whether," and, well, "work" again bind these lines more closely together with repeated sound. And togetherness, after all, is what this poem is all about. It makes total sense that a message of connectedness would be delivered with so many connected sounds echoing all over the place. It looks—and sounds—like we're not alone in the universe after all.