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.
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