Throughout this poem, the city of Chicago is personified as a strong, resilient working man, "a tall bold slugger" with "Big Shoulders," "laughing," "half-naked, sweating, (and) proud." The city laughs in the face of adversity. For example, in the third stanza, the line, "Under the smoke, dust all over his mouth" possibly alludes to the Great Chicago Fire of 1871, which destroyed over three square miles of the city, killed approximately three hundred citizens and left thousands more homeless. The city of Chicago survived the fire and got on with the project of "Building, breaking, rebuilding," and it laughed while it was rebuilding "with white teeth," implying that even so great a disaster as the fire was no match for its strength and resilience. In this stanza of the poem, Sandberg references the defiant laughter of the city eight times.
This brings us to the specific lines you have asked about. In these lines, Chicago, personified, brags and laughs because it has "the pulse" under "his wrist." The pulse refers, I think, to the collective pulse or heartbeat of the citizens of the city of Chicago. The suggestion is that the personified figure of Chicago is powered by the collective pulse or heartbeat of all of the city's citizens—and he laughs because he is so sure of the strength of that collective pulse. The people of Chicago have strong hearts, which allows the city to face any challenges with confidence. In the previous line, the city is compared to an undefeated "fighter" who laughs because he "has never lost a battle." The laughter is a sort of "Bragging" laughter, the kind of laugh Muhammad Ali might have laughed if challenged to a fight by Woody Allen.
The phrase "the pulse" may also have another meaning, signified by the word "the." Sandberg might have written "a pulse," but the fact that he wrote instead "the pulse" perhaps alludes to the position of Chicago at the beginning of the twentieth century, when the poem was written. Chicago was an agricultural and industrial powerhouse of the country, providing the country with livestock ("Hog Butcher for the world"), wheat ("Stacker of Wheat"), railway lines and engines ("the Nation's Freight Handler"). So perhaps the city of Chicago is "Bragging" because under its wrist is the pulse of the whole nation. It provides for and feeds the rest of the country.
When Sandberg writes that Chicago had also "under his ribs the heart of the people," this is a continuation of the meaning behind the image of the pulse under the wrist. Chicago is personified throughout the poem as a huge, strong figure ("Big Shoulders," "tall bold," "Fierce," "Shoveling"), which in turn implies that the heart beneath its ribs is also huge and strong. And as the heart is commonly a symbol of inner strength and vitality, the implication is that the people of Chicago are strong, enduring, vital people.
The next line in the poem consists only of one word, "Laughing!" This, the sixth of eight references to the city's laughter in the third stanza, again re-enforces the impression of Chicago as a confident, defiant, and proud city, laughing in the face of troubles, secure in its own strength and vitality. The importance of this laughter to Sandberg's vision of Chicago is emphasized by the fact that "Laughing!" is given its own line, by the fact that the preceding comma isolates it in its own clause, and also by the fact that is followed by an exclamation mark.