I measure the voltage ESC drawing while increasing the dc motor speed. Multimeter shows that as long as the speed increases the voltage value decreases. Can anybody explain why this is happening?
You can't understand this because you don't know what you observe. The electromotive force (EMF) of your motor is proportional to its speed (the ratio is called Kv). So, when the speed of your motor rises, the EMF rises too.
What does an ESC ? Basically, it hashes the input voltage to generate the expected output voltage.
So, what is this decreasing voltage value you observe ? It won't be the output voltage of the ESC, because it's rising, not falling, and, as it is hashed, you probably can't measure it with your multimeter (basic multimeters only measure continuous or sinusoidal signals).
I expect the decreasing voltage to be the input of your ESC. Why would it drop ? Because your driving a motor with a load that rises with speed (typically a propeller), and this requires current. The wires between your battery and your ESC are resistors. Let say you have 50mOhms of resistivity. If you uses 10A, the input voltage of your ESC is 0.5V less than your battery output. Moreover, your battery also have internal resistivity. If you drive more current, the output voltage of your battery drops too.
I think the answer here should be insightful. But basically when the motor is running, you're working under the equation
This equation has to stay balanced, so assuming your (P)ower remains a fixed value, then if (V) changes, current (I) has to make and opposite change in order to keep the equation balanced. In this instance, current (I) increases as speed increases, thus you get a related decrease in (V)oltage in order to maintain the balance.
EDIT: My answer is flawed because as pointed out in the comments, the assumption of power being fixed is wrong and doesn't make any sense.