Life for them is one continuous grind. Day after day they go through the same monotonous round of duties, while those that are "getting along" are using their initiative to get greater fullness of life. There is nothing so responsible for poverty as this lack of initiative, this power to think and do for ourselves.
Does get along mean to succed here and why is this phrase in quotes?
Here, "getting along" means "making progress", "having some success" in life and/or business. I think the phrase has been put in quotes to mark that it's being used in an informal, idiomatic way. It could be read as: "... while those that are getting along (as they say) are using their initiative..".
Note that "getting along" often has the different meaning of having a good, smooth relationship with someone. "My boss and I are getting along much better since I started arriving on time".
not a teacher