"The Splendour Falls on Castle Walls" is partly a description of a beautiful scene in the mountains and the speaker's response to the echoes of a bugle. Listening to those echoes, though, makes the speaker reflect on different kinds of "echoes." What kind of "echo" has the past left on the present? Does ancient mythology leave a kind of "echo" in the modern era? And—perhaps most importantly for our speaker—what kind of an "echo" will we leave behind us when we're gone… gone… gone…? (See what we did there?)
Ready for a downer? The various kinds of sound "echoes" in the poem (alliteration, internal rhyme, end rhyme, repetition) help to emphasize the poet's anxiety that all of his poetry is only empty noise and sound, like an echo.
Think the past is over and done? Nope. The "echoes" described in the poem are the literal version of the kind of figurative echoes that the past can have on the present.