Why was "The Love Song of J. Alfred Prufrock" a major literary breakthrough in poetry?

T. S. Eliot's "The Love Song of J. Alfred Prufrock" is a poem which brought verse fully into the twentieth century as it effected a cultural shift in poetry from Romanticism to Modernism.
Eliot's poem is also the first English-language poem of the twentieth century that is composed of free verse, and it differs greatly from the poems of the Romanticists because in them the natural world is in sympathy with man and offers succor, whereas for Prufrock in Eliot's verse the images of nature are sick or ominous. In the first stanza, for instance, the evening "spreads out against the sky" like "a patient etherised upon a table." Further in the poem, the atmosphere of the city is ominous, even suggestive of evil in its imagery, as there is "yellow fog" with "yellow smoke" on the window-panes. 
Indeed, Eliot became a major voice in Modernism as he has expressed so well the lassitude of man, as well as the horror and incongruity of many aspects of modern life. But, unlike Ezra Pound, who felt that the poet was not obliged to recompose the world out of its fragmentation, but could, instead, elicit "a paradisal aspect" out of such disjointedness, Eliot perceived a necessity to establish order out of the fragmentation through re-composition. In other words, Eliot wished to seam the world together as a spiritual quest.
Editors Richard Ellman and Robert O'Clair write that Eliot was strongly influenced by the French symbolists, while the metaphysical poets, too, affected Eliot strongly: 

His principal models were Laforgue and Corbiere. He dealt almost exclusively with decadent, enervated people, yet in all his technical devices revealed a violent, innovative energy. He combined a precise and often formal outward manner with an inner writhing, bound together by wit.

J. Alfred Prufrock is the "decadent, enervated" man of the twentieth century. Moreover, he exemplifies modern man so well because he shares the angst of many. For instance, his inner fears prevent him from risking rejection by a woman despite his strong sexual desires. Much like characters brought to life by James Joyce, his inner dialogue is wrought with fear and a certain spiritual paralysis. Prufrock, too, is afraid to engage with the world:

Is it perfume from a dressThat makes me so digress?Arms that lie along a table, or wrap about a shawl.    And should I then presume?    And how should I begin?

Despite the frustrations and hesitations and fragmentation, Eliot's verse, like the writings of other Modernists, has a profound honesty to it.
Additional Source: The Norton Anthology of Modern Poetry