Why do weights store more energy when a weightlifter lifts them up from the ground?

You need to provide energy to lift the weight. If the weight falls or otherwise goes down, the energy is freed again - for example, it can drive a motor for a short while. It is therefore convenient to consider that there is stored energy in the lifted weight. More accurately, the energy is stored in the relation between the two attracting objects - in this case, planet Earth, and the weight.