Is this sentence grammatically correct? This quote shows that Charlie understands that her mother will be happier there, so she will have to deal with the satiation.
I think you meant “situation” and not “satiation” in the last word. The word “satiation” means satisfaction, which is not something you would normally “deal with” as in your sentence.
The sentence is technically grammatically correct. It is a compound-complex sentence with two independent clauses and a subordinate clause.
Independent clause 1: This quote shows that Charlie understands
Subordinate clause: that her mother will be happier there
Independent clause 2: she will have to deal with the situation.
It is important to be clear when writing an essay. No matter who the audience is, it needs to be evident what you are talking about.
Your understanding of your audience...is integral in determining the development and organization of your argument, as well as the stylistic techniques you can utilize in your writing. (enotes)
Although there is no context for this sentence, there are some things that could be done to it to make it clear. First, who is the “she” and who is the “her” in the last clause? It is not completely clear whether we are talking about Charlie or her mother who has to deal with it. One can assume it is Charlie, since the mother will be happier, but it is not clear. Would her mother be happier if she dealt with the situation? If you take one or both pronouns and replace them with the noun they are referring to, your sentence will be clearer.