Why does sitting on soft sofas cause back pain?

Soft = less support. If a soft sofa increases your pain, then it may be related to the fact that it provides you less support. Specifically, in a sitting position, it may be allowing your spine to flex (arc forward) more than if you were sitting upon a firm chair, for example. This is classic for disc related pain. An evaluation with a physiatrist or other spine specialist can help evaluate your specific symptoms.
Bad springs. Old sofas or couches, as well as old mattresses, can cause back pains due to bad springs.