To measure the current flowing through a wire, you need to insert your ammeter in the circuit and have the current flow through it. That means you lose some voltage on the ammeter and the less you lose, the less it will influence your measurement. So you want your ammeter to have as low a resistance as possible.
To measure the voltage between two points, you would want to have _as little current as possible_ flow through your voltmeter, to cause as little of a voltage drop as possible. So you would want your voltmeter to have as much resistance as possible.
In both case the point is to minimize the change to the thing you are measuring by measuring it.
You don’t want your measuring device to change the thing you’re measuring. Since series resistances add together, having your ammeter add as little resistance as possible minimizes its influence on the current.
Likewise, changing the current flow will change the voltage drop (in most circuits). Since the voltmeter is going to span between two different voltages, current is going to want to flow, and resistance is a measure of how difficult that is.
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