Why didn't cheese spoil when it was given as rations to soldiers?

Firstly, I would guess cheese is a food made to last but the more important point is that modern ideas of what is spoiled and what is acceptable to eat are probably quite different than the way people felt about it 100 years ago. People did not have effective refrigerators and food was carried by slower transportation. So things like brown spots on bananas were no big deal and actually spoiled parts of a fruit (or cheese) that you could not or would not want to eat were simply cut out. Even moguls like Rockefeller probably found acceptable things that a modern person (in the USA, anyway) would toss.

Dry hard cheeses like parmesan generally do not go bad quickly. As the FDA puts it *"As a general rule, hard cheeses such as cheddar, processed cheeses (American), and both block and grated Parmesan do not require refrigeration for safety..." Such cheeses are in fact typically aged in warm environments for months or (in the case of really fancy cheese) years. Mold can also be scraped off of the cheese, leaving the rest edible. Certain types of mold are simply left in the cheese to add flavor even today. The cheese would likely have been stored in large blocks, slowing spoilage greatly and of course as others have said, probably wouldnt have been carried around in a backpack for months at a time.

*https://usdasearch.usda.gov/search?utf8=%E2%9C%93&affiliate=fsis&query=cheese&commit.x=0&commit.y=0&commit=Search

...the first of these I filled with provisions—viz. bread, rice, three Dutch cheeses, five pieces of dried goat’s flesh (which we lived much upon), and a little remainder of European corn, which had been laid by for some fowls which we brought to sea with us, but the fowls were killed. There had been some barley and wheat together; but, to my great disappointment, I found afterwards that the rats had eaten or spoiled it all. As for liquors, I found several, cases of bottles belonging to our skipper, in which were some cordial waters; and, in all, about five or six gallons of rack.

(Robinson Crusoe)

They allowed the salty young cheese or any of the hard sorts get dry. Before 20th cent no plastic bags existed, and to keep it moist for long time specially would be considered as a crazy idea. So, they carried with themselves very-very dry and hard cheese. These cheaper young sorts are not fat. And it is fat that makes the dry cheese go spoiled so fast.

Notice also, that people BC did not travel really for long without any access to the food. The fast march when they couldn't stop to buy or rob anything could take days, but not months. The furthest non-supply travels were these in desert. And when everything was OK, it took maximally several days. Notice, that armies did not cross Sahara, and small troops can use oasis's sources. And when an army had to cross a desert, then problems began, as for Alexander before and after India.

As for Medieval times, Europeans did travel even less these times. Mongols killed antelopes or horses and put meat under the saddle - to be by spiced by horse sweat. Had they invented the method? I don't know.

I don't know , when the Medieval times end for you, but in 15-18 cent they traveled far by ships. And I don't remember any mention of cheese as a common mariner's food. For officers only. Salty dried meat/fish and dried bread or biscuits, left for some time or beaten against the table to make the worms crawl or fall out - that was the food. For officers, too. What is funny, according to the contemporary dietology, they should eat their bisquits together with insects in them :-)

First and foremost, for an army recommendations and even regulations would be always conditional on availability; if there was not enough cheese available or if cheese had gone bad then simply cheese would not be distributed. In other words, the fact that cheese was recommended does not mean that every time the soldiers had to do long marches they would have been supplied with it.

Second, I think you imagine a marching army that carries with itself all the supplies needed for the march/campaign. More often than not, the army would forage or resupply on the march, through the purchase of food (if in friendly terrain), the use of previously established supply depots¹ and by simply plundering whatever they needed (if in enemy territory or if the locals were not willing to provide food)².

An army trying to march with all of its supplies has lots of complications; not only you need a long supply train that you need to defend and that slows down the army, but you also need to feed the people and the animals of the supply train itself, thus putting an upper limit to its utility.

Obviously, since one of the main functions of these depots was to store foodstuff, they were designed in order to conserve it as best as possible. In "The Logistics of the Roman Army At War", by Jonathan Roth, it is stated (p.185):

The ancients had the technology to store grain for as long as ten years. The Romans understood that in order to preserve grain for the longest possible period, the temperature and moisture content of the grain had to be kept as low as possible

And, while cereals were the main aliment for the soldiers, such depots contained other items (p.187):

A granary or horreum did not only store grain: the low temperature and ventilation in granaries helped to preserve fresh and salted meat, cheese, lard, vegetables, olive oil and wine. Frontinus refers to the “food supplies” (alimenta) stored in the horrea of the Roman army after the battle of Teutoburgerwald in 9 A.D.230 Several horrea, such as those at Balbuildy and Ilkley in Britain, have revealed amphora fragments. The author of the African War refers to a camp storing, in addition to grain, wine, and oil, “other necessary items which had been gathered as provisions.”

Additionally, when available, carrying supplies by ship (either through the sea or through rivers) allowed them to be transported faster.

And let's not forget that it was not only the army that didn't have access to fridges. Conservation of the food was an important issue to everybody, so long lasting varieties would be preferred for storage. You would get to eat fresh cheese (or meat) only shortly after it was prepared; everything else would be elaborated/salted/smoked so it could last as long as possible.

²In page 200 of Jonathan Roth's book, "The Logistics of the Roman Army At War", we find an upper bound for how far an army could go from its supply base:

In practice, the Romans did supply armies overland for well over 100 km. (60 miles)—and occasionally up to 320 km. (200 miles). There are a number of examples of this attested in the sources.