What do Marx and Engels mean as they write, "the proletarians have nothing to lose but their chains?"
To understand this quote and how stirring it can be we need to understand the basic Marxist principle of class and how he presented a vision of the world where the working class was dominated by the upper class who controlled society and kept all power for themselves. As a result, the working class were left to face and confront massive inequalities and hardships that were never faced by the upper class or aristocracy. Marx and Engels therefore believed that the only future for such a situation was for the working class, who were infinitely more numerous than the upper class, to revolt against the aristocracy and create a new communist society that was based on common ownership and equality.
Therefore, if we go back to your quote for one moment, given the dire situation facing the working class, otherwise known as the proletariat, and the lack of power that they have, they have nothing to lose by revolting, as their situation is so dire anyway. The only thing they can lose if they are successful are the chains that so tightly bind them to a position of complete inferiority and hardship. This is how we can explain the quote you have given in relation to key ideas of Marxism.