As reported from the NOAD (New Oxford American Dictionary), that is used to replace who, whom, and which; it is also used to introduce a defining or restrictive clause, especially one essential to identification.
The book that I've just written.
In the sentence I'm really happy with the insurance company that helped me with my claim, helped me with my claim identifies the company you are happy with; you are not happy with every companies. The same is true for I am forever indebted to the charities that helped me.