IANAL and you shouldn't get legal advice from a website. But in my opinion there are only two possible reasons to care about licensing:
You want to use the code. You need the copyright holder's permission, and thus you must use it under whatever license the copyright holder(s) gives you. In this case you get to choose because it is dual-licensed. It is a logical contradiction to license the software simultaneously under both licenses at the same time because the GPL adds requirements that the MIT license does not. Therefore you as the user can choose one OR the other. Note: if you are distributing the code you can presumably carry-forward the dual-licensing too if your own project is dual-licensed accordingly. But if you want to operate under the terms of just one license you can.
You want to contribute to the code. In that case you need to know that your code will also be dual-licensed. For example, it is generally impossible to contribute to the Linux kernel any code that isn't GPLv2. And if you ONLY want to contribute GPL code (because you are a strong believer in copyleft) then you must be aware that jQuery is GPL AND MIT, which means your code must be similarly GPL and MIT.
