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.
Triethanolamine is a pH adjuster. It is a toxic component that can be absorbed by the skin and detected in blood later. It can be a dermal irritant. Though it's considered safe, you should think before using it in skincare. There are many products on the market without Triethanolamine.
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.