The speaker of "Apollo Musagetes" begins by describing Mount Etna, the fiery volcano that he is climbing. (The volcano appears in other of Arnold's poems as well.)
He then warns that Apollo (the Greek/Roman god of the sun and music) does not spend time here, but rather at a much calmer place where the cliff meets the sea. At the top of that cliff, flocks of sheep and pigeons roost, while shepherds sleep in the hills, wrapped in blankets.
Suddenly, several forms appear in the moonlight, clothed in white and singing - it is Apollo and his nine muses. All of them are "divine," though Apollo is most stunning. Apollo and the muses bathe in the spring on the cliff before heading up to Olympus, their godly abode. On their way, they sing about what has been and always will be, about the father of all things, about the other gods, and about the actions of men.
Finally, they sing about nature — about day, night, and the stars.
This poem is an excerpt from Arnold's 1852 verse drama Empedocles on Etna, but has remained much more well-known than the larger work. The same is true of his poem "From the Hymn of Empedocles," though the context of that work is important here. The verse drama follows Empedocles, a man famous for his stoicism, as he battles the powers of the intellect and the passions on his way to throw himself into Mount Etna for the thrill of the experience.
Apollo is an important figure both in the larger work and here in the poem. The classical god of the sun and music, Apollo is known both for his nine muses - the goddesses who inspire artists - and for his association with man's rational powers. For the Greeks, Apollo represented man's ability to construct artwork from experience, whereas other gods - most notable Bacchus, who figures into Arnold's poem "Bacchanalia" - represented the irrational tumult of experience itself.
Thus, this poem is an articulation of the powers of the sublime and beautiful. The speaker has chosen to live amongst Etna, described through violent phrases like "black, rushing smoke-bursts." This setting stands in stark contrast to the pastoral landscape that Apollo frequents. Just like in "A Dream," Arnold juxtaposes water and fire in this poem, describing the fiery Mount Etna as fierce and destructive, and the cliff by the sea as serene and beautiful.
Like in many of his poems, Arnold's speaker envies the ability to live a serene life. While this speaker is mired in blackness and dynamic danger, he watches (or imagines) the possibilities than an uninterrupted intellect (as reflected by Apollo) might bring. He dreams of the insight such a perspective would inspire, reflected by the songs Apollo and his muses sing about everything including man and nature. Because they can rest calmly on the cliff, they learn songs to celebrate everything. In a way, Apollo can be likened to the Scholar-Gipsy of that poem, a figure who has refused to life amongst the turmoil of desire and ugliness, and thereby learned more about himself and the world while on his way to the grandeur of Olympus.
Read in context of Arnold's work, the poem can also be seen as another critique on society, albeit a subtler one than usual. While the support of the pastoral life is entirely unambiguous, it is notable that the mention of songs about "the actions of men" carries a subtle hint of dissatisfaction. Whereas the immortals know "rest," men know "action," the type of action that has led the speaker onto the fiery cliffs of Etna. However, there is a hint of optimism, in that the speaker does not rest in this dissatisfaction, but instead ends the poem reflecting again on nature, on its potential to lead us to somewhere greater than the world we otherwise know.