Why in an inconsistent axiom system every statement is true? (For Dummies)

I'll try to say this all in plain English:

Let's say we decide to accept the following two facts: (1) "I am a fish", and (2) "I am not a fish". Just keep those in mind.

Now let's pick any old statement, say: (3) "You can fly". Now let's prove that the statement is true!

Alright, we've already accepted that (1) "I am a fish". Of course, any time I have a true statement P, I can make a new true statement by making the statement "P or Q is true." Because to check if an 'or' statement is true, I only need to check that one of them is true. (If I tell you "My name is Dylan OR I can spit fire," you don't need to wait around with a fire extinguisher to tell if that statement is true. It's true because the first part of it is true).

So by this logic, the statement (4) "I am a fish or you can fly" must be true (since the first part is true.)

OK, but now let's say, in general, I have some 'or' statement "P or Q" and I know for a fact that the whole statement is true. If I also know that P is false then I can conclude that Q is true. Right? Because an 'or' statement is true if and only if at least one of the statements inside it is true, so if I rule out one of them the other one must be true. (So if I always tell the truth and I tell you that you have a billion dollars in your bank account OR I just ate a sandwich, you can check your bank account and quickly conclude that I just ate lunch... unless you're very wealthy.)

Alright, so far so good. We know the statement "I am a fish or you can fly" is definitely true. But wait, we also know that the statement "I am a fish" is false (remember, it's one of the things we assumed in the very beginning!). So that means, by what we just talked about, that the statement "You can fly" must be true.

So voilà! Using the magic of a contradictory system, we've proven you can fly!

It is somewhat misleading to say "every statement is true" about an inconsistent theory. This might be a point of confusion in this question, and in general the difference between "truth" and "provability" causes many other confusions, so we have to be careful to distinguish them.

"Truth" is a property that a statement has in a particular model. In other words a particular statement is either true or false in a particular model, assuming the statement is written in a formal language compatible with the model.

An inconsistent theory has no models at all. If you have no models, there's no model in which any particular statement can be true. It is correct to say that every statement in the language of the theory is provable from an inconsistent theory, and that every statement in the language of the theory is semantically entailed by the theory. But it's an abuse of language to say that every statement is "true" in the theory: true in what model?

Sometimes, when people are writing an informal proof, they say things like "assume $A$ is true" or "assume $B$ is false". But these are just figures of speech; the actual proof system usually has other ways of dealing with hypotheses than to mark them as "true" and "false". Alternatively, you can view those sayings as abbreviated forms of "assume $A$ is true in some fixed, unspecified model", "assume $B$ is false in our fixed, unspecified model". That interpretation of the informal proof is fine for any consistent theory. But that interpretation is more difficult for inconsistent theories. Because there are no models, it will be a counterfactual statement.

The answers in the prior question all employ syntactic consequence, i.e. consequence by proof in some formal system. It may be clearer to instead consider this as a semantic consequence, i.e. $\rm\ S\models Q\ $ means that $\rm\:Q\:$ is true in every model ("possible world") where every member of $\rm\:S\:$ is true. Thus $\rm\ \{P, \lnot P\}\models Q\ $ is true vacuously, since there are no models where both $\rm\:P\:$ and $\rm\:\lnot P\:$ are both true, i.e. there is no model witnessing a counterexample to that consequence, where every element of $\rm\:S\:$ is true but $\rm\:Q\:$ is false. You can find some further discussion in the Wikipedia article on this Principle of Explosion.