The speaker says that the “Brain” is "wider" than the "Sky" itself, and then asks the reader to imagine holding the two next to each other in order to compare them.
Literally, of course, the human brain is much smaller than the sky (and it's impossible to arrange either "side by side"!). But the speaker is really talking about the human mind, and saying that its capacity to perceive and imagine feels as vast and expansive as the sky above.
The capitalization of "Brain" and "Sky" is in keeping with Dickinson's unique poetic style, but also might help clue readers into the fact that the speaker is considering both of these entities symbolically, as representative not just of the literal brain and literal sky, but of curiosity, wonder, imagination. The direct repetition, or diacope, of “side by side” reinforces the similarity between the mind and the open sky, placing them next to each other as equals. The assonance of these lines reinforces the connection as well: note the long, slow /i/ sounds in “wider,” “Sky,” and “side by side,” which seem to expand the lines themselves.
The caesuras here, in the form of dashes, draw out these lines even further—in other words, they make the lines "wider"! Dashes, like idiosyncratic punctuation, are another characteristic of Dickinson's poetry. Here, they not only "widen" the lines themselves, but also suggest a burst of energy at the end of lines. Instead of ending with an expected, final-feeling period, the speaker uses dashes at the end of nearly every line in the poem to capture the potential of the human mind; they move the reader forward through the poem, creating a sense of spaciousness and momentum.
Finally, these lines establish the poem's ballad meter, which alternates between iambic tetrameter and iambic trimeter. An iamb is a poetic unit with two syllables, the first of which is unstressed and the second of which is stressed: da-DUM. Tetrameter means there are four of these iambs in a line, making eight syllables total, while trimeter means there are three iambs per line, for six syllables total:
The Brain—is wider than the Sky—
For—put them side by side—
Dickinson used this meter quite often in her poetry. It has a bouncy, predictable rhythm that evokes religious hymns.