Water is a great solvent. It’s neutral and doesn’t provide any strong positive or negative effects. It just helps components to mix better and to transport active components.
Cocamidopropyl Betaine is a pH 4.5-5.5 liquid that is used to improve the overall efficiency of the formulation. It’s usually combined with different surfactants and is used because of its cleansing efficiency. It helps to remove sebum and dirt from the skin and hair. It also works as a foaming agent helping to make a pleasant foam.
Polysorbate 20 doesn’t have any notable skin benefits but is used in cosmetics quite often. Manufacturers use it for formulation purposes. It improves the scent, texture, and feel of the product.
PEG-40 Hydrogenated Castor Oil helps ingredients to mix better. It has nice cleansing activity helping to remove sebum and dirt from the surface of your skin. It's also responsible for making soft foam in soaps, shower gels, etc. Despite it having "PEG" in the title, it's considered to be safe.