Any measurement of a circuit disturbs the circuit in some way. So, if the input Z is purely resistive and it's 10 Megohms, it will have negligible effect on measuring the voltage across a motor whose internal resistance is 2 ohms. 2 ohms || 10 Meg ohms is pretty close to 2 ohms. The || means in parallel with.
If your circuit had a resistance of 10 G ohms and you put a 10 M-ohm resistor across it, you can't measure it with your meter. 10 G ohms || 10 M ohms is closer to 10 Meg ohms than 10 G ohms and the voltage measured will be nowhere near right.
A DC measurement that you make will suck some current out of the circuit just to measure it and it can be quite small. It just so happens that a 10 Meg ohm input R is a standard number, say for a handheld DVM. With Lab instruments it may change based on range, getting higher with the more sensitive scales.