Washing soaps are made by a chemical reaction of oils and fats. They are mixed with alkali to form soaps such as sodium hydroxide. In this process, the acids which are present in oil get mixed and start converting into their particular salts. Alkali is usually added in order to neutralize fatty acids.
It is also seen that in order to make washing soap wet, a large amount of sodium silicate is also added. The alkali which is present in soap is usually hygroscopic in nature. Hygroscopic is a process in which substances are able to absorb water molecules through absorption.
So, when the rainy season starts the alkali component which is present in washing soap becomes hygroscopic which leads to precipitation of fatty acids. This is why soaps become soggy.