I'm not familiar with the term "handling frequencies." A natural frequency, however, is simply the frequency at which a structure responds to a force. For example, when someone jumps off a diving board, you can watch the end of the board flex up and down at maybe 3 or 4 Hz.
Now if you were to try to shake the board using a constant force starting slowly then speeding up, you'd find that there wouldn't be a whole lot of response until you get close to that natural frequency. At this point, the deflection of the tip of the board maybe several times what it would be if you just held a constant force on it. But if you keep going faster, it'll die down again. Keep going faster still and you might find another natural frequency where the board wants to bend in an S shape rather than just deflect at the tip, etc, etc.
This is important because, as ICT mentioned, autopilot systems and especially fly by wire control systems might try to respond to a disruption such as turbulence by rapidly commanding control inputs. On an airplane with a lot of flex, that's like shaking that diving board. So they put in notch filters that reduce the amount that the autopilot is allowed to move the controls at frequencies near the natural modes. But as these modes are known through analysis and ground testing, accelerometers are not usually required.
However, some larger aircraft might use wingtip and/or tail mounted accels to measure fatigue cycles or even try to minimize wing flex. A good example of that is the C-5 which has an active lift distribution system that keeps the wings from flapping too much.