This poem begins with a flat, dull description of a flat, dull day. Take a look at the way the speaker uses parallelism to introduce an uninspiring landscape:
The Sky is low — the Clouds are mean.
Those mirrored words ("the ___ is ___") suggest that, everywhere the speaker looks, the world seems so gloomy that there's no point in even varying one's sentence structure to describe it. Everything's just one big samey mass of gray.
Not only is the weather lousy, but it also seems actively unhappy. Personifying the sky as "low" and the clouds as "mean," the speaker suggests that nature itself is in a bad mood, both depressed and sour.
And that bad mood will turn out to be exactly the subject of this poem. It's not just that the weather is bad outside: it's that the speaker's internal weather is "low" and "mean" as the skies. This will be a poem about how the outer world can seem to reflect the inner world.
This flavor of personification was so common in 19th-century poetry that one important artist and critic, John Ruskin, coined a dismissive term for bad versions of it: the "pathetic fallacy," the attribution of human feelings to nature. ("Pathetic" here doesn't mean "pitiful," but "to do with emotion.") Anyone who's seen a movie in which a storm breaks out just as something sad happens to the hero will be familiar with how the pathetic fallacy works. Ruskin (and many thinkers who followed him) disliked this technique because he felt it was often hokey, false, and trite.
But Ruskin would likely have approved of Dickinson's personification here. In this poem, the speaker is well aware that they're projecting their own feelings onto the landscape, and actively reflecting on the human tendency to do so!