When one says "Allah's essence", what does it actually mean?

The essence that is being referred to here is in Arabic called الذات (Dhaat), and the attribute is called الصفة (Sifah).   Let's say that one of your names is Aaminah (the one who believes). This name refers to your essence, meaning that it refers to you as an individual. It also refers to the attribute contained in the name, which is Iman (belief).  So, now you are attributed/characterized with the attribute/character of belief because that is what your name means. Therefore, the difference between essence and attribute is that essence is the individual and attribute is the characteristic/description of that individual.

And Allah knows best.