Simply put, in the timeline of life, winter comes before spring. As soon as winter is finished, that means that spring is coming next, not far behind. For the entire poem, Shelley has been talking about the west wind, and all that it does. He describes in detail the times that the west wind blows, its effect on the earth and on people, and his feelings towards it. Often, the descriptions are tumultuos and ominous; for example, the west wind is a bringer of "rain and lightning," "black...hail", "storms", "fear", "tremble", and "chasms" in the sea. So, the west wind, for much of the poem, is described as a fearful thing, bringing storms, hail and destruction. But, in the very last stanza, he ends on a hopeful note. He says that despite all of this, the west wind brings in the winter. And, what is at the end of winter? Spring. So, even though the wind brings cold, stormy winter in, there is spring at the end of it to look forward to. And in this sense, Shelley calls the wind a prophet, giving "prophecy" (a prophet often is able to tell the future); the prophecy that the west wind "trumpets" to the world is that spring will be coming soon.
I hope that those explanations help a bit; good luck!