"Others" could generally refer to other people—that is, everyone who isn't Shakespeare. But more specifically, it could refer to other poets, other writers like Matthew Arnold himself. "Others abide our question," says the speaker—in other words, other writers are willing to be questioned and to provide answers to those questions in their art. But Shakespeare does not need to, for in his work he displays what Keats famously referred to as "negative capability"—that is, the remarkable ability to explore the complexities of life without putting forward a precise philosophy or system of ideas. Shakespeare simply stands back and lets his characters speak; he offers no solutions to life's myriad problems, no neat resolutions of thorny moral issues.
Lesser mortals—Arnold among them—may continue to "ask and ask," to probe the Bard's works in search of a point of view, but they will always come away with more questions than answers, awed and beguiled in equal measure by the "immortal spirit" that is Shakespeare and the inexhaustible complexity of his artistic vision.