The force of the US military is not what props up the value of the USD, look at all the other currencies around the world with almost no offensive military. How do you explain the value of CHF or JPY? The value of a currency is based on the network effect of everyone who uses it and how much they trust it not to devalue (ie trusting they won't print too much too fast).
We trust the USD because everyone we have ever met in the US has trusted it for their entire life, and their parents, and grandparents, ... . And the network effect is so large that people growing up in countries with weak currencies trusted it more than their home currency, it was stable and everywhere.
All other fiat currencies are similarly backed by a government that enjoys monopoly of legal force over some geographical area. That's why when governments die (lost war, revolution) the value of their fiat goes to zero and only "money" with intrinsic value is accepted (gold, silver, gems, works of art, cigarettes). In 6000 years of recorded history money always either had intrinsic value or was backed by a strong guy.
We trust the USD because everyone we have ever met in the US has trusted it for their entire life, and their parents, and grandparents, ... . And the network effect is so large that people growing up in countries with weak currencies trusted it more than their home currency, it was stable and everywhere.