Anti-essentialism is a Popperian idea that many people are either unaware of or do not understand. Many people are essentialists, particularly people who think they understand philosophy, but essentialism is a bad mistake. There are two separate ideas that Popper criticises. (1) Essentialism is the idea that reality consists of ultimate essences and we ought to try to explain what we see in terms of ultimate essences. (2) There is another closely connected idea: we ought to define our terms before we start a discussion otherwise we might get lost.
Let’s take point (1) first. Suppose that reality does consist of ultimate essences. Whatever they are we don’t have direct access to them and so the idea that we should use them seems to require knowledge we don’t have. What would an explanation in terms of essences look like? We start with terms like “cat” and then define the essence of a cat by listing all the features that all cats have in common: whiskers, weird looking eyes, make meowing noises and so on. We would then take all of the cat features and use them to explain what cats do. The problem is that each time we define an essence we use many undefined terms and so we would have to define the new undefined terms and we would get into an infinite regress without ever explaining anything. Nor can definitions reduce ambiguity, as mentioned in point (2): every definition we introduce uses undefined ambiguous terms.
Popper suggests that a better way of thinking about definitions is that a defined terms should be used as shorthand for a longer description: methodological nominalism. So instead of saying “negatively charged particle with spin-1/2 and about 1/1000 the mass of a proton” we say “electron” as a shorthand.
Furthermore, if we try to explain things in terms of ultimate essences we might be tempted to think the ideas we have tell us what the essences are and that would be bad because we might be wrong. An example of this: I have seen some philosophical discussions in which the participants start the discussion by defining knowledge as justified true belief and discussing that definition. The discussion didn’t go anywhere because there was nowhere for it to go: the problem had been set up in such a way that it was completely unsolvable.
One last comment I should make that I don’t think was made by Popper, although I could be wrong. It seems to me there’s another reason to reject essentialist methods: it takes explanations and breaks them up into pieces that are difficult to understand. Suppose we want to understand electrostatics: the forces between slowly moving charges. We find that sometimes two objects attract one another and sometimes they repel one another. We propose and test the idea that the force with which they do this varies as the inverse square of the distance between them. We also discover that the force between two objects isn’t determined by the distance between them and introduce different charges on the objects to explain this. Imagine trying to learn this by having a dictionary that defines charges, and distances between charges and forces and so on. You would have to try to take all these different definitions and put the information they give together in an order that would actually give you an explanation before you could understand electrostatics. You can’t start with an unexplained heap of definitions and then use them to work out a theory. You have to start with problems and explanations.
The best expositions of Popper’s anti-essentialism that I have been able to find are in The Open Society and Its Enemies, Vol. 2, Chapter 11, Section II and Conjectures and Refutations, Chapter 3, Section 3.