Instead of an anchor, imagine a lump of very dense material, where the density is 1000 kg/L. This is a 1000 times as dense as water, and a cube of this material with a length/width/height of 10 cm would weigh a tonne.

So the cube is in the boat, pushing the boat down and displacing 1 tonne of water (1 m^3).

You throw the cube off the boat and the boat rises, because it doesn't need to displace so much water. The weight now displaces hardly any water at all (actually 0.001m^3).