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what does it mean when an anghle is equal to itself?


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It is just another fancy way of saying "You are as old as you are," or "You are as tall as you are."  No matter how old or how tall you are, you will always be the same age and height as yourself, now won't you?

Basically, it's like looking into a standard mirror (not a funhouse mirror) and seeing you, who will always be equal to you.

It is the reflexive property.  Anything that is equal to itself is reflexive.  It is not so much a=b because there is only one variable.  So it is simply a=a, like looking in a mirror.

We all know that any number is equal to itself. For angles, what we mean is that it is congruent to itself. More generally, any object is the same as itself, which is is consistent with the definition of sameness. In all cases, this is known as the reflexive property, which, along with symmetry (a=b implies b=a) and transitivity (a=b and b=c implies a=c) form the definition of equivalence.
This sounds like a statement from a geometry proof, right? Proofs often make very obvious statements called "identities" as part of the very formal nature of the proof. Usually, saying that"something is equal to itself" is a set up to add that something to something else. For example, if I say that: a=b and angle3=angle3, then a+angle3 must = b+angle3 because "equal things added to equal things result in equal things" Hope this helps.