Why does the speaker have tears in his eyes when he hears the west wind?

The west wind in the poem “comes from the west lands,” and so does the speaker.  So whenever the west wind blows, the speaker is reminded of his home, and the tears are caused by homesickness.  The speaker has worked hard, and is weary of his toil and his life, a claim substantiated in the second verse, with the line “It’s a fine land, the west land, for hearts as tired as mine.”  He remembers it as a gentle, green place, “where men may lie at rest.”  The speaker longs for such rest, and longs for home.  The rest of the poem paints a picture of the speaker’s homeland, and when he hears the west wind, he hears it calling to him:  “Will you not come home, brother? Ye have been long away.”  The wind repeats this supplication several times, until the speaker decides he must return home, to a place where he may find “sleep for aching eyes.” 

So, upon hearing the west wind, the speaker is reminded so strongly of his home that it moves him to tears, and compels him to take action.  He has been away from home for a very long time, and with the whispering of this warm wind he makes the decision to leave the hard life he has been living and rediscover, as he states, “the land where I belong.”