Why can't ice transfer coldness into another substance?
Because nothing can transfer 'coldness' to anything else.
However, ice can remove thermal energy (heat) from other things. This is why we use it to cool our drinks. Thermal energy transfer is at a fundamental level very simple. Energy moves from where there is more to where there is less (which is why 'coldness' cannot be transferred. As ice melts, energy is transferred from the surroundings to it, and so the surroundings become cooler.