Peace loli.
My quick understanding (without having researched the verses in-depth).
The Qur'an, in its entirety, does espouse itself as a manual on how to not only create a balanced individual life but also a balanced society. With that context in mind, it becomes clear that this verse is about society. The verse refers to kinsmen, orphans, travellers, the needy, as well as God and the Messenger.
Why is that important? One could ask, like you have done, how one gives physical possessions/money to God. We can take this even further and ask how we can give physical possession/money to the Messenger, who is dead. Using common sense (the Qur'an urges us to use our intellect), a very reasonable interpretation is that we are to give this "booty" to what God and the Messenger stand for, i.e. the cause of God. What is the cause of God? Simple. It is what I mentioned in the first paragraph: it is a just society that is balanced and that serves its members like a well-oiled machine.
We see the phrase "God and the Messenger" uttered many times throughout the Qur'an, especially in regard to following God and the Messenger, and the question can also be asked in reference to those verses: how do I follow the Messenger, who is no longer alive. It is simple. You follow God and the Messenger by following the Message with us today. In connection with this verse, what does the Message tell us? It tells us to help the needy, the traveller, etc. Basically, it tells us how to become a better society. By trying to accomplish those ideals, we are in fact obeying God and the Messenger. We are in fact giving the "booty" to the God and the Messenger, in a very real way.
Peace.
