mutable modifier in this context is used to indicate that a data field of a const object may be modified without affecting the externally-visible state of the object.

If you are thinking about caching a result of expensive computation, you should probably use this keyword.

If you have a lock (for example, std::unique_lock) data field which is locked and unlocked inside a const method, this keyword is also what you could use.

You should not use this keyword to break logical const-ness of an object.

Example with caching:

class pi_calculator {
 public:
     double get_pi() const {
         if (pi_calculated) {
             return pi;
         } else {
             double new_pi = 0;
             for (int i = 0; i < 1000000000; ++i) {
                 // some calculation to refine new_pi
             }
             // note: if pi and pi_calculated were not mutable, we would get an error from a compiler
             // because in a const method we can not change a non-mutable field
             pi = new_pi;
             pi_calculated = true;
             return pi;
         }
     }
 private:
     mutable bool pi_calculated = false;
     mutable double pi = 0;
};

By default, the implicit operator() of a lambda is const. This disallows performing non-const operations on the lambda. In order to allow modifying members, a lambda may be marked mutable, which makes the implicit operator() non-const:

int a = 0;

auto bad_counter = [a] {
    return a++;   // error: operator() is const
                  // cannot modify members
};

auto good_counter = [a]() mutable {
    return a++;  // OK
}

good_counter(); // 0
good_counter(); // 1
good_counter(); // 2