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.

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_.