This poem is about the transition from Winter into Spring. But the opening line also indicates a transition from night to day. 

The light comes brighter from the east; the caw

Of restive crows is sharper on the ear. 

The sun rises in the east. Therefore, the sunrise is brighter than the sunset. Paralleling the seasons, perhaps the speaker also indicates that the sun is brighter (more full of life) in the Spring than in late Autumn or Winter. Conflating sunrise and the Spring, we have natural symbols of the beginnings of life. Roethke was deeply in tune with nature, very much like the Romantic poets in England and the Transcendentalists (such as Emerson and Whitman) in America. Roethke once said he could sense the moods of nature. 

The theme of the poem seems to center around things coming to life. The sun rises and our senses are sharpened as we become more conscious (awake). The sun starts to defrost the earth. The snow drops from the trees and the cold roots begin to stir. The mood is very positive. Nature is waking up and opening itself to the world. Roethke, being mentally in tune with nature, compares this awakening and "opening up" in nature with a conscious awakening or flourishing of the imagination in his mind: 

And soon a branch, part of a hidden scene,

The leafy mind, that long was tightly furled,

Will turn its private substance into green,

And young shoots spread upon our inner world. 

The branch is part of a "hidden scene" because it was hidden under snow and is now budding. This notion of "hidden" also refers to the hidden mental realm of the speaker. The pun on "mind" opening up describes the actual leaf opening and describes a person's mind waking up (back to the awakening of sunrise and spring) and engaging with the world (nature). The speaker personifies the leaf, giving it a "mind" further establishing the connection between himself and nature. 

Our "inner world" is that world of nature before it opens up and this also refers to our inner, mental world. This establishes the connection between the awakening in nature and in the mind. This also establishes the idea that as nature opens itself to us, we open ourselves to nature. This is the kind of connection, between humans and nature, that Roethke was interested in and pursued in his poetry. The extended metaphor, by the end of the poem, works both ways. The awakening of nature is a metaphor of the awakening of a person to nature; and vice versa. Thus, in the last line, "shoots" can be roots underground and they can be thoughts in the mind.