I have just learnt the basics of Bayes Theorem.
I did want to search the below myself instead of asking on Maths Stack Exchange, however I have no idea what this is called, or even trying to show me.
Could someone explain what this is called, what it is trying to show, and please help me understand it?
For this answer, I will avoid a measure-theoretic, rigorous treatment of probability given the nature of the question.
In probability, you are concerned with likelihood of outcomes. Let $ \Omega $ denote the set of all possible outcomes of an experiment. Concretely, take the experiment of rolling a 6 sided die. The set $ \Omega $ in this instance is the set containing the possible outcomes of the experiment. Each outcome can be represented by the number face up after the roll. This gives $ \Omega = \{ 1, 2, 3, 4, 5, 6 \}$.
An event is a subset of the sample space. For example, the event that you roll an even number is $ E_{\text{even}} = \{ 2, 4, 6 \} $. The event that you roll a 5 is $ E_5 = \{ 5 \} $.
In general, a partition of a set is a collection of subsets that cover the set (their union is the entire set), in which no two sets intersect.
So, a partition of the die roll example may be the events $\{2, 4, 6\}$ and $\{1, 3, 5\}$. This is a partition because the events cover the space: the die roll is either even or odd, and the events do not intersect: the die roll cannot be both even and odd.
Note that, for example, the events $ \{ 2 \} $ and $\{2, 4, 5\}$ are not a partition.
This should be enough for you to follow the reading, noting that $ \vee $ is the symbol for 'or', $ \wedge $ is the symbol for 'and', and $ | $ is the symbol for 'conditioned on'.