Lines 1-5 begin to establish the poem's voice and setting. The speaker is an artist reflecting on his "craft" or "art," which he pursues in the silence of nighttime. Since there is no additional or alternative explanation, it makes sense to assume that the speaker is Dylan Thomas himself, reflecting on the art of poetry. A poem that reflects on the poet's art is called an ars poetica (Latin for "Art of Poetry"), after a famous poem of that name by the ancient Roman writer Horace.
Thomas wavers slightly between calling poetry a "craft" and an "art," implying that it might be either or both. A "craft" sounds slightly less prestigious than an "art"; though it implies skill, it's a word often associated with homemade or hand-made goods. This initial ambiguity sets up a tension that runs throughout the poem: Thomas takes his poetry seriously, but resists taking it too seriously or coming off as pompous. He wants to keep his poetry humble and honest, not chase literary prestige. In short, he seems to aspire to the humility (but not the amateurishness) of a craftsman, as well as the seriousness (but not the elitism) of an artist.
Thomas also describes his art with a curious adjective: "sullen." This word usually means sulky or resentful, but its meaning here is somewhat mysterious—and likely metaphorical. In light of the lines that follow, it might imply that poets "Exercise" (i.e., practice) their art away from the rest of humanity, as if sulking in their rooms. In other words, while "lovers lie abed," poets stay up all "night," writing in solitude. "Sullen" also hints that Thomas's art is rebelling against something. The rest of the stanza will provide some clues as to what that might be.
The poet practices his art while "only the moon rages," somewhere in the sky above. This detail seems highly symbolic: the moon is conventionally associated with passion, romance, poetry, and the like. Its "rag[ing]" might indicate something about its appearance; perhaps the moon is reddish (red being the traditional color of anger) or surrounded by stormy weather. But the verb "rages" might also be meant to personify the moon and to project some of the poet's own feelings onto it. In the "still[ness]" of the "night," that is, the poet wrestles with raging, tumultuous emotions, which seem to find a mirror in the moon overhead.
Meanwhile, the "lovers" of the world rest quietly "abed" (in bed). Here, Thomas turns a potentially sensual image into a sad one. Rather than making love or sharing pillow talk, these couples, according to Thomas, "lie [...] With all their griefs in their arms." In other words, they lie together in deep, brooding sorrow, metaphorically clutching their "griefs" as tightly as they embrace each other. Unhappy as they are, these lovers seem to provide a starting point for the poet's "art."