The reason for multimeters to have a high input impedance
It's not that multimeters have a high impedance - just voltmeters do.

When measuring voltage, you connect the voltmeter (a multimeter in voltmeter mode) to the nodes between which you want to measure: it is a parallel connection. In order to measure correctly, you need the impedance of the voltmeter to be as high as possible so that very little current goes through it, so that it doesn't disturb the circuit (the current that was going through the, for example, resistance that you are measuring voltage on will continue to be almost exactly the same, thus the voltage drop will as well).

When measuring current, on the other hand, you connect the ammeter in series. This time, to prevent the device from disturbing the measuring, its impedance should be as low as possible, so that the circuit keeps behaving the same way (no voltage drops in the ammeter and the current flowing in that branch of the circuit stays the same).