The poem opens with a paradox that makes a bold and intriguing statement: a lot of what is considered "madness" is, in fact, "divinest Sense." By opening in such a way, the poem immediately calls of both these polar opposite categories into question, asking the reader to re-examine the definitions of "madness" and "sense" (or insanity and sanity). The poem isn't necessarily focusing just on literal sanity and insanity in a clinical sense, but rather on what the "Majority" of people (as mentioned in line 4) think is normal. This is a poem about outsiders (like Dickinson herself) as much as it is about actual madness.
The alliterative /m/ in "Much Madness" establishes that there is a lot of what is considered madness out there in the world—but that, likewise, much of it is misunderstood. Not only is this "madness" frequently misunderstood, it's actually "divinest Sense." Here, sense means something like insight or forward-thinking. The "sense" possessed by some people who get mischaracterized as mad is sometimes the most profound thought of all. A good example might be a writer who is way ahead of their time, like Dickinson herself was. Though she is now considered one of the divinest—as in, best—poets of all time, she was misunderstood in her own day.
The word "divinest" also has connotations of godliness, suggesting that some of these instances of "madness" are actually revelations of the highest order. Note that Dickinson is far from the only poet to draw a link between so-called madness and visionary power—William Blake, William Wordsworth, and Samuel Coleridge all draw similar parallels particularly in their thoughts about poetry.
Line 2 helps put the first line into context. It states that the ability to look beyond received ideas of "madness" and "sense" depends upon a "discerning Eye." That is, it takes a certain kind of unflinching wisdom to look at the world clearly and question the status quo. This idea can be applied to numerous moments in human history, which is part of the poem's power. For example, those who pushed for the abolition of slavery had the ability to "discern" that it was morally wrong, and fought against the "majority" position. Likewise with the feminist movements of the 19th and 20th centuries. The "Eye" is not specifically about the visual sense; it is a metonymic way of referring to anyone who has the ability to tee the world afresh, and not believe things are a certain way just because that's what everybody else thinks.