Perhaps we could explain it using an example; see if I truly understand this:
Lets say we are in a lift, the lift has just started accelerating upwards at 9.8m/s2. Now, the Lift is a non-inertial frame of reference, because it is constantly accelerating. But, we look at the inside of the lift. It is not moving relative to us inside it, so we believe it is stationary. We are taking it to be inertial, but it is not.
Now, we realise to our horror that we weigh twice as much as we should, shown by the weighing scales that are for some reason in our lift with us. Since we think the lift is not moving, we think that the increase of apparent weight is an increase in gravitational pull.
No we look out of the window in the lift and see the shaft whizzing past, we realise we are moving and accelerating, so we take it to be non-inertial, and can attribute our increased weight to the lift accelerating upwards and into our feet, pushing us up.
I think that explains at least some aspects of it, I am sure it is far more complex.