The novel is full of characters who are only completely unable to communicate with one another. What's more, most of them don't really even have a way of internally processing the events happening around them. Some, like Bounderby, are in total denial, while others, like Louisa, are too deeply detached from their emotions to react appropriately to anything. In order to really show this, Dickens uses a narrator who is able to explain to the reader what each character is feeling regardless of whether that character knows this information or not.
Remember, for instance, the scene when Gradgrind is asking Louisa if she wants to marry Bounderby. There is a really tear-jerky moment when the narrator tells us that if only Gradgrind hadn't built up a huge wall around all of his emotions, he would have seen that Louisa was ready to talk to him about all her doubts about this kind of marriage. At the same time, if only Louisa could see that her father really did care about her underneath the Fact-based exterior, she would have started to cry on his shoulder. Instead, all we get of this enormous missed opportunity is the narrator telling us how it could have been.
At the same time, the very removed narrator stance is also helpful when Dickens is making fun of the action. Think about the way Bounderby's bombastic speeches are usually surrounded by the sarcastic remarks by the narrator. The third person narrator is also used to play up the weepy factor. We see this when the narrator describes small emotionally resonant moments like Louisa staring into the fire for hours on end, or Rachael and Stephen walking together down the street.