Why an accelerometer shows zero force while in free-fall I'm trying to explain the effect in the title a non-physics person (of chemistry background) and I found it very hard to explain in simple, intuitive ways. I'm not a physicist either, so following the famous quote If you can't explain it simply, you don't understand it well enough, I decided to ask :)
Background
I was playing with a 3-axis MEMS accelerometer on a device and I had developed a PC program to visualise different sensor values in real time. The accelerometer proved very fun to play with - as you are rotating the device around, the gravity-induced force transfers between the three axes, but also responding to motions, etc.
Then I tried throwing the device in the air and noticed that while the device was in flight, all 3 axes showed zero - as if the connection was off and I was reading nothing. In fact, the connection was fine and the values I read were very close, but not exactly zero. Then I thought - "but of course, in free fall you don't feel any acceleration, so the accelerometer shows zero". This is the part I find hard to explain to the other person - why the device has force acting on it (gravity), but the sensor inside "sees" no force applied.
An example of an intuitive type of answer I was trying to construct are round soap bubbles - they become squished while resting on a surface (same as an accelerometer laying on a table shows 1g in the relevant axis), but when allowed to fall freely, they become round, no squishing (0g on the accelerometer). But the soap bubbles experience a ton of air resistance, as well as wobbling, so it isn't a very good example.
Considerations
I'm trying to get an explanation that doesn't involve reference frames, as we both don't have an intuition about them. Please excuse me if I'm not using the right terms, as I said I'm not well-versed in physics.
 A: Imagine that you are standing in a train and have roller skates on.  
If the train is at rest you can stand in the train without having to grab hold of anything.  
If the train starts to accelerate in the forward direction, what do you have to do to stop yourself moving in the backward direction?
You need to hold on to something and have the train apply a force on you in the forward direction to keep you up with the train.
That force applied on you will increase your speed so that you keep up with the speed of the train.
The larger the acceleration of the train the larger the force which needs to be applied on you.
The train now reaches a constant velocity, what do you have to do to stop yourself moving in the backward direction?
Nothing at all as you can stand in the train without having to grab hold of anything.
This is the phone being thrown in the air and the accelerometer registering an acceleration of zero which I will explain later.
If the train starts to slow down, what do you have to do to stop yourself moving relative to the train?
You need to hold on to something and have the train apply a force on you to keep up with the train.
This time the force will be in the backward direction as that force will be trying to slow you down to move at the same speed as the train.  
A lot of accelerators work this way.
You have two objects with one of the objects having a much smaller mass than the other.  
The smaller mass is almost completely free to move other than it is coupled via a flexible link which exerts a force when distorted.  
So here is you acting as the small mass to measure the acceleration of the train.  
 
The spring will exert a force on you so that you (almost) keep up with the train.
With the train and you  at rest or moving with constant velocity the distance $AB$ will stay the same. 
With the train accelerating the spring will be compressed and exert a forward force on you and the distance $AB$ will increase.
With the train slowing down the spring will be stretched and exert a backward force on you and the distance $AB$ will decrease.
So the distance $AB$ is a measure of the acceleration of the train.  
Note that when the "reading" of acceleration is taken there is actually no movement between the train and what is actually been measured is the force that is exerted on you for you to keep up with the train.
So you might measure the distance $AB$ and use Hooke's law to work out the force which is being exerted on you, place a parallel plate capacitor between $AB$ and measure the capacitance of the capacitor which can then be related to the separation of the plates and hence the change in distance $AB$ or you can push or pull a piezoelectric crystal which will produce a voltage which depends on the amount of push or pull.  

The accelerometer in your phone will be able to sense the accelerator in three dimensions and will probably be of the change in capacitance type as shown below.  

So why do you get a zero reading on your phone when it is in free fall when up in the air?
It is because the sensor is measuring the acceleration between the phone and the moveable "you on roller skates" within the phone and as the phone and you are not accelerating relative to one another the relative acceleration (which it is measuring) reading is zero.  

For details of the inner working of real sensors this article is a good start?
A: There are many kinds of accelerometers, but all have one thing in common: They have a test mass that is somewhat free to move. The accelerometer measures the motion (or attempted motion) of this test mass. Whether the accelerometer case is accelerating but the test mass isn't, or the test mass is accelerating but the accelerometer case isn't, the accelerometer will sense this because the test mass is accelerating with respect to the case.
But the accelerometer will register no acceleration if both the accelerometer case and the test mass undergo the same acceleration. Any force that acts identically on the case and on the test mass results in an acceleration that is undetectable by the accelerometer. This leads to the Newtonian explanation of why an accelerometer cannot sense gravity. The Earth is extremely large compared to an accelerometer. At the scale of an accelerometer, the Earth's gravitational field looks very much like a uniform gravitational field. The test mass and accelerometer case undergo almost exactly the same gravitational acceleration and hence this acceleration cannot be detected by the accelerometer.
The relativistic explanation is even easier. Gravitation is not a real force in general relativity. Gravitation is no more measurable by a local measuring device than are the fictitious centrifugal and Coriolis effects.
A: Since you've asked for an explanation without involving reference frames, I'll base my explanation on more familiar concepts of force and acceleration.
A typical accelerometer could be modeled as a mass on a spring. The position of the mass, when the spring is not compressed or stretched, is defined as zero and corresponds to zero reported acceleration.
When the spring is compressed or stretched, the displacement of the mass from its zero position is measured (using one of many available techniques) and reported as a positive or negative acceleration.
So, what is reported is a displacement of the mass, which may or may not correspond to the acceleration of the accelerometer that we would measure externally.
The diagrams below illustrate how it works in x or y directions, where gravity does not play a role. 

Whenever the body of the accelerometer accelerates, it pushes or pulls the spring, which, in turn, pushes or pulls the mass. The force required to accelerate the mass, ma, will cause the spring to compress or stretch, and the resulting displacement of the mass will be measured and reported as a positive or a negative acceleration.
For z direction, illustrated on the diagrams below, the dynamics are different.

At rest, the spring is compressed or stretched to counter the weight of the ball.
In a free fall, when all parts of the accelerometer experience the same acceleration, the mass accelerates due the gravity - not due to the push or the pull of the spring. Therefore, the spring does not experience any forces, hence, there is no compression or stretching and no displacement, hence, the reported acceleration is zero. 
This description would not be entirely accurate, if the resistance of the air was taken into account. Since such resistance would slightly reduce the acceleration of the falling accelerometer, the spring wold have to compress or stretch a bit to adjust (reduce) the acceleration of the mass accordingly.  
