In a nutshell, ingredients with emulsifying properties let two things which normally repel, come nicely together - in skincare this is normally oil and water. They do this by reducing the surface tension between these ingredients, letting them mix and form a great textured product.
If emulsifiers were not included in cosmetics products, you’d see lots of separation between the “incompatible” ingredients. Think of what happens to balsamic vinegar when you add it to olive oil, it sits separately with the balsamic making dark, delicious blobs in the olive oil. If you added an emulsifier, the olive oil and balsamic would mix together as the new ingredient would lower the surface tension between the oil and balsamic. Unlike our olive oil and balsamic mix, in skincare, most of the time we want a nicely blended products that have a lovely consistency. The emulsifier helps us to achieve this by bringing together the normally incompatible ingredients.
You’ll also read about “surfactants” in skincare and they sound an awful lot like they do the same thing as emulsifiers. Both reduce the surface tension between substances that normally repel each other - so have the same overall purpose. The main difference is how they are used (with emulsifiers forming a particular sub-class of surfactants with a special purpose). Surfactants are (generally) used in cleansing products - they lower the surface tension between oil and water to allow oil and dirt to be washed away in water. In contrast, emulsifiers are really for creams and lotions that need to bring together both water- and oil- based ingredients. It’s a bit confusing because they really are two words to describe the same thing but used for different purposes - one uses its powers for cleaning; the other for mixing.