What is the narrator trying to express when he thinks "I should have been a pair of ragged claws/ Scuttling across the floors of silent seas"?
All of what has been said in previous answers is correct; however, noting the stream-of-consciousness form used in this poem, the line must be taken as Prufrock's internal aside, a thought he has to himself as he considers his options. He wants to bring up his "overwhelming question," yet he knows not how to introduce his ideas into the conversation. He fears, as is seen in his previous worries about how people view him, with his bald spot and scrawny legs, how his thoughts will be recieved. He must think his question important because he thinks it has the ability to sum up the universe:
To have squeezed the universe into a ball
To roll it toward some overwhelming question,
To say: "I am Lazarus, come from the dead,
Come back to tell you all, I shall tell you all"—
Still, he fears that presenting his thoughts in this social circle will bring him ridicule. He fears more time "pinned and wriggling on the wall" where he will be scrutinized and critcized for his differing opinons. So rather than face the contempt of his peers, he backs away from the question, much like a crab walks backwards, and remains silent. Then he thinks to himself that no matter how grand his thoughts or intellect, he might as well be a crab (and note the synecdoche that makes him even less than a crab) since he has no courage to speak his mind. It is really very sad.
A professor of mine in grad school insisted that much of this poem should be read with humor—lines such “shall I wear my trousers rolled” and “do I dare eat a peach” provide good examples of this. The humor would be based on irony, the speaker in the poem not as aware as his creator or we the audience of what a pathetic figure he is. This line can be read in the same spirit: on the one hand it is, as other responders have explained, very poetic, using devices such as synecdoche to create the image of the crab, which is the speaker, while the sibilants in “scuttling,” “floors,” “silent” and “seas” invite us to slide over, if not linger on, these words, all interrupted with the harsh “k” and “t” sounds of “scuttling,” almost onomatopoetic in their effect. But then the humor: this guy wants to be a crab; he thinks he’s as low as a crab—his self image is really that bad? Do we laugh at him or pity him?
You are not alone being confused by these lines. Scholars have long debated the lines. Some believe the reference is to a lobster, but most to a crab because the emphasis in the line in on "scuttling." The technique Eliot uses is synedoche, substituting the claws for the crab. Eliot is known for his literary allusions. The poem draws heavily on a connection to Shakespeare's Hamlet. The crab imagery is most likely an allusion to a comment Hamlet makes to Polonius in Act II: "For you yourself, sir, should be as old as I am--if, like a crab, you could go backward." Hamlet is contrasting his age and virility with Polonius old age. Prufrock similarly seems to wish that he could have moved backward to relive the opportunities he missed. As with any Eliot line, one reading is not enough to understand the layers of meaning he includes. The claws are also a likely continuation of the sexual imagery throughout the poem connected to fingers, hands, and arms.
The speaker of Eliot's poem is expressing his sense of isolation and loneliness, and his inability to fit into a world that he perceives as hostile. Better for him, he thinks, to be a creature like a crab who is protected by a thick outer armor and better still, that he does not have to truly interact with anyone or care about what they think of him, or he of them.
"Just love the poem", as I have, life-long.... It gets better, over seventy: no profs to please, and the mermaids are beginning to sing to me!
Don't overthink it. That's the point of the line. The claws are disembodied, mindless objects moving senselessly through peaceful surroundings. No responsibilities, people or distractions -so no neurosis.