The speaker begins by mentioning the "sun-clouds," a phrase that creates the image of sunlight bouncing off (or perhaps filling) the clouds in the sky. These clouds, the speaker says, "cannot manage such skirts." This is a difficult line to understand, since it's not immediately clear what the speaker means by "skirts." However, the title of the poem provides some clarity, hinting that the speaker is comparing the color of the clouds to the color of nearby poppy flowers.
In this context, the word "skirt" refers to the outer edge of something, as the speaker describes the petals that create a colorful rim around the center of the poppies. More commonly, though, the word "skirt" refers to the lower half of a woman's dress or gown. This subtly associates the poppies with womanhood, establishing the poem's (admittedly ambiguous) interest in femininity.
The speaker goes on in lines 2 and 3 to refer to a woman bleeding in an ambulance. It's unclear who this woman is or why she's in an ambulance, but she's bleeding enough that the blood seeps through her coat. The speaker seems oddly undisturbed by this grisly sight, saying that the woman's heart "blooms" through her clothing "astoundingly"—a description that romanticizes the woman's pain, almost making it seem as if she's a flower in bloom. The startling color of the woman's blood still pales in comparison to the vividness of the poppies, however. In other words, even the most extraordinary sights in the speaker's life are at odds with the image of the poppies.
From the poem's very first line, the speaker makes the environment of the poem feel especially alive by personifying the sky as something that actively competes with the poppies. This personification makes it seem as if everything in the speaker's surrounding world is trying, and failing, to measure up to these beautiful, late-blooming flowers. This juxtaposes the poppies' beauty with the rest of the world, which the speaker presents as comparatively drab, even if the sights the speaker mentions are actually quite remarkable—a sign that the speaker is unhappy or unimpressed by most things in life.
"Poppies in October" is very ambiguous, so it's hard to pin down and make sense of its potential references. However, it's possible that the speaker's interest in poppies is an allusion to the British holiday known as Remembrance Day, when people display paper or cloth poppies in November to commemorate soldiers who have died in battle. This has been in practice since just after World War I, and the holiday is even known as "Poppy Day." Even if the poem isn't explicitly about Remembrance Day, the association between poppies and this holiday imbues the poem with thoughts of death—something that is clearly on the speaker's mind, given the morbid fascination the speaker seems to have with the woman bleeding out of her chest.