As is frequently the case in his work, Tennyson gives us the sense of an irresistible power in nature in the opening of this famous poem. If the first stanza is looked at in isolation, the meaning is one in which a dichotomy is set up between the force residing in the sea and the weakness and vulnerability of the speaker, whose tongue cannot articulate his thoughts.
No stanza that is only a part of a given poem can be examined totally in isolation. It's conceivable, however, that Tennyson could have ended "Break, Break, Break" after only the first four lines, because this contrast between natural—or by extension, divine—power and human frailty is a complete thought in itself. It is the flip side of the description of nature at the start of "The Lady of Shallot," in which the serenity, the benign rural scene, is ironic in the context of the tragedy that will be enacted:
On either side the river lie
Long fields of barley and of rye,
That clothe the wold and meet the sky ...
The sea is the opposite of the peaceful (and man-made) fields. Though the speaker views the ocean's force in negative, remote terms ("thy cold, gray stones"), it becomes clear that it is only to him that it is in some sense alien.
As we read on, what is the significance of the other humans who come into play—the "fisherman's boy" and the "sailor lad" singing? The speaker feels remote from not only the force of nature, but from other people who interact with it and seem to revel in it. His expression is dominated by a sense of loss of the "vanished hand" he will never again touch and the voice he will never hear.
But what does this loss have to do with the indomitable sea? The contrast with his own grief can be seen in both negative and positive terms. If the sea is characterized by its cold gray stones and crags, it also indicates something that goes on in perpetuity. It is an abode (to paraphrase Shelley in "Adonais") where the eternal is, and thus, it also suggests that the loss the speaker has endured can perhaps be remedied by the idea of eternal life symbolized by the sea.