It seems that the world, or objective reality, obeys logic ... logic pretty much is how the world works

I am going to give the non-idealist (non-Kantian) answer, and say that what you are talking about is whether reality, in some sense, conforms to logic. (Or, the other way around: we invented a formal system of logic that, at its core, just reflects reality.)

Well, we might think of it this way:

Syntactically, logic is just a game of little symbols. If you see a P, and a Q, and a little arrow in between, and then see another P, then you can write a Q. Those are just the rules of the game.

However, the semantics of our logic is one of truth. The semantic interpretation is as follows: If the premises are true, then the conclusion is also true. Keep in mind that we can do "logic" without ever talking about the concept of "truth": namely, when it's just a little symbol game.

When we add notions of "truth" to our logic, we give it meaning. It is the semantics to our syntax.

Now, think of geometry. Back in the day, we only had Euclidean geometry. Nowadays, we have non-Euclidean geometry. The reason why non-Euclidean geometries weren't popular back in the day was not because people were too dumb back then and couldn't create geometries that were logically consistent and didn't include Euclid's parallel postulate. Rather, it was because there seemed to be no use for it. It was just ... a symbol game, since there was no "model" or semantics for it. Originally, people thought that Euclidean geometry described the world, so its "meaning" or "model" was essentially the universe or reality. That's why it was #1.

However, after Einstein, we learned that the universe actually isn't Euclidean: that the actual structure of space violates Euclid's parallel postulate. What this did was give a serious boost to non-Euclidean geometries, since now we know that Euclidean geometry wasn't actually instantiated or modeled by the universe, or reality. Conversely, now we had a model for non-Euclidean geometries. Non-Euclideans stopped just being mental gymnastics and symbol games. This breathed life and meaning into them.

So, away from geometry and back to logic. Our semantics or "model" of logic is given in terms of truth. But what is "truth"? Many would say that "truth" is just that which represents (corresponds) with reality. Some statement is true if and only if it holds in reality. "The cat is on the mat" is true if and only if the cat is actually on the mat. So, reality is the model of logic.

If that's the case, then logic really is a matter of the "world" and "reality", as you suggested. Let's think about it intuitively, too. Classical logic says that A is A, A or ~A and ~(A and ~A). Now let's think of reality. If something is the case, then it is the case. If I see a ball, the ball is the ball. Secondly, either this is the ball, or it isn't the ball. And finally, it's never that something is the ball and isn't the ball at the same time. You've always observed these things to be true about the world around you.

So, to answer your question, logic "works so well" because it is a true reflection of reality. It's just our formalization of reality. Inversely, I guess we can say, as you did, that "objective reality obeys logic".