If the premises are inconsistent then they cannot all be true. So the statement "if all the premises are true, then the consequence is true" holds, since the premises are never all true.
On the syntactic side, the details depend on what deductive system we use, and there are many different possibilities even if we confine ourselves to natural deduction-style systems. In the kind I prefer, there is a primitive falsity sentence $\bot$ and negation is not primitive, rather $\lnot A$ abbreviates $A\to \bot.$ So clearly from $A$ and $\lnot A,$ you can use implication elimination, aka modus ponens to infer $\bot.$ The principle of explosion is encapsulated in an elimination rule for $\bot$ that says you can any sentence from $\bot$.
It might seem unsatisfying that explosion seems to be assumed, not proven, in this framework but this is actually a major advantage, especially if you’re particularly interested in the rule. This way it is simple to track what proofs “use explosion” and which don’t, and it’s easy to come up with a systems (e.g. minimal logic) that don’t have it: just consider subsystems that don’t have this rule. The question of “why” it’s true classically, is better answered by a semantic argument, anyway, e.g. the one I gave above or observing $A\land\lnot A\to B$ is a tautology.
Regarding your last paragraph, it doesn't matter that you can ignore the contradictions in your premises. If you can find a consistent subset $\Sigma'$ of your assumptions $\Sigma$ such that $\Sigma' \models \phi,$ that's great, and much more interesting that the fact that $\Sigma\models \phi.$ But that just means you should have been working with $\Sigma'$ all along rather than $\Sigma.$
We know that since $\Sigma$ is inconsistent, $\Sigma\models \psi$ for any sentence $\psi,$ so the fact that $\Sigma\models \phi$ is not very informative. Thus we only usually care about reasoning from a consistent set of premises. (Unless, say, we’re using a logic that doesn’t have explosion, where the consequences of an inconsistent set of premises can be nontrivial.)