Why multiply a matrix with its transpose? This might be a very stupid question, but I do not seem to understand why I would multiple a matrix with its transpose. I am not a mathematician, but I am very interested in understanding the practical usage of equations: 

Imagine I have three products sales Apple, Orange and Pear for the last 3 days in a matrix form called A: 
$$ A= 
\begin{bmatrix}
        Apple &amp; Orange &amp; Pear \\
        10 &amp; 2 &amp; 5 \\
        5 &amp; 3 &amp; 10 \\
        4 &amp; 3 &amp; 2 \\
        5 &amp; 10 &amp; 5 \\
        \end{bmatrix}$$

What will $AA^{\rm T}$ tell me? 

I have seen this long answer link: Is a matrix multiplied with its transpose something special?, but I did not get it at all. 

I see that a lot of equations use the product $AA^{\rm T}$ and I really hope that someone will give a very simple answer.
In your case, $AA^T$ just sitting on a park bench doesn't tell you anything of great interest.

Hyprfrcb's answer talks about units. One of the elements has units of squared apples. Another has units of pear-oranges. Another has units of orange-pears! This by itself should be a red flag that values don't mean anything by themselves.

It can be a means to get to a least-squares solution if you were looking to model, say, how much of each fruit you'd expect to sell on a particular day. (This was one of the answers in the linked question.)

But by itself? It's just a fruit hybridization experiment gone terribly wrong.
I've been trying to figure this out my myself recently, and I think I understand it.  

John is right, in your example, it doesn't make sense.  But I'll attempt to explain it in a simpler example than what Brethlosze said... (bare with me and please tell me where I am wrong).

Let's say you are measuring something like acceleration.  But your measurement of acceleration always has an error.  Each time you measure acceleration 'a', you have an error of +- 0.1*a.

This error propagates as you continue your calculations.  If you want to calculate velocity from that acceleration, then that acceleration error is going to propagate into the velocity calculations.  If you continue to calculate distance, than that acceleration error is going to also continue to propagate into the distance calculation.  

Now this is a very simple example, but it shows the relationship between this measurement (and its error) and all the calculates values (state variables).  Now imagine a more complicated scenario where you need more than 1 measurement to get the desired state variable that you are looking for.  You will have an equation with 2 variables to show the relationship between the measurements and the state variable.  What if you are looking for 4 state variable that each are composed of 2 other state variables that those in itself take 2 or 3 or 4 or however many measurements to calculate.  Now you can use a matrix to show the relationships between all these measurements and state variables.  

So now, if we transpose the matrix and multiply it by the original matrix, look at how those equations in the matrix are being multiplied with all the other variables (and itself).  Try the math of a simple 2x2 times the transpose of the 2x2.  This is the covariance.  Wikipedia: In probability theory and statistics, covariance is a measure of the joint variability of two random variables.  Too me, its like the covariance matrix mixes all the variables and measurements in every way possible to show how all of them vary against and with each other.  

This is very important in inertial measurement units (IMUs).  There are many error states (sometimes more than a dozen) in an IMU in order to calculate the state variables (position, velocity, body orientation, etc.).  These error states and state variables are defined in a matrix.  When you find the covariance of the matrix, you can define the joint variability between all these error states and state variables.  This is important in order to know how these errors and measurements can affect your accuracy of position, velocity, and body orientation.  By knowing your covariance, you pretty know how accurate your IMU is at defining your position, velocity, orientation, etc.