Otherwise, it doesn't sound so bad. But it's probably way more profound and deep than what I'm putting on it by taking it at face value.
tl;dr: It's a call to arms that the proletariat should rise up to break their chains from the economic system that
creates the bourgeois. That while the bourgeois have all the money, they'll always be dependent on a proletariat that allows this to continue.
As a socialist, I agree with the sentiment... In the form of unions, civil rights, democracy and the enabling thereof, social programs targeted at the poor and those in the lowest income tax brackets, sweeping federal programs that improve the livelihoods of all, collective ownership of the social machines that produce currency, et cetera. Where I disagree with Karl Marx's Manifesto is that he requires a revolution be fought that essentially decapitates the head of society and throws it in a bloody heap through violent revolution. The next step thereafter is the institution of a dictatorship run by a few who would eliminate all remaining private enterprise, and then dissolve to create a society where everything that exists is owned by all who live within it. Ergo the word: "Communism." Literally "communal ownership of all by all for the benefit of all." It's a nice sentiment, and a great idea, until you put people into the picture.
And then you get millions of dead Ukrainians and a madman tyrant who murders his own son in the name of an ideology that he warped to give himself absolute power.
So the significance of the quote is that it's a call to literally overthrow the business class and the rich at
any cost. Including human life. "You have nothing to lose but your chains" implies the assumed fact that the proletariat have nothing, and so there's nowhere to go but up. It was written in a different time than we live in now. It compares the proletariat to serfs working the fields of rich nobles in the medieval era. That's what it means.
So next time a hobo stabs you for change, he's breaking free of his chains by ripping you down and taking what you have.
YEAH! DEATH TO WHITEY!
FEAR THE RISE OF THE PROLETARIAT!
Funny, a white man wrote the book.