Disclaimer: most of what I'm about to write applies to 1300's Hungary, especially the details of how land ownership works. Be warned ye Anglophiles.

Government as a separate entity didn't really exist here - you had a king, who had some royal holdings and some nobles that (hopefully) listened to his orders and had holdings of their own. Roads were either on the land of some nobles, or were a part of royal holdings, and whoever had them as his property had to pay for their upkeep.

This upkeep was a massive sum - a castle needs repairs, servants and soldiers need to be paid one way or another, shields need replacing constantly and so on. Even roads need to be maintained by someone at times, and bridges are even more problematic. Then there's river trade, which has docks (wood in water needs replacing) and lost ships.

Next are administrative expenses. As a noble landowner, you are required to pass judgement on matters there, and these need to be recorded properly, which means scribes, archives and parchment, ink and quills. None of that is exactly free.

Then there's police force, or rather a force of armed men to keep order. These could be mercenaries or poorer nobles (familiares), regardless, you need to pay them. They do buy equipment for themselves, but they do it with the money you pay them, so you're effectively paying for that too, especially if you consider that if you want a better armed force, you need to pay them more just for them to be able to buy the equipment.

Next is development of your land - as any avid gamer can tell you, you need to keep an eye on your eco at all times. These aren't always expenses per se, they often took the shape of rulers attracting people into a particular village by giving it tax breaks for a number of years, expecting it to flourish and return investment a decade or so down the line. That said, some villages did get payment from the local noble on founding, usually taking form of things like livestock, wood (often just a permit to cut a number of trees), tools or necessities (e.g. salt).

Then comes a seldom mentioned part of lavish lifestyle - patronage. You basically pay scientists and artists you have in your court to do their own thing, and the more you're willing to spend, the bigger the prestige you get.

And last thing I'll mention is investment into trade - you give a merchant 50 gold, he gives you a piece of parchment promising you to pay you back 60 gold, or give you some form of property or another. That exploration expense you mentioned above is just this trade investment, only on a grander scale - you expect (or hope that) the newly discovered lands that you have trade routes to will bring in a tidy profit.

With all that, it's not that surprising that nobles often gave their castles to other nobles because they borrowed money from them. This style of holding a castle for a period of time, until profits from it pay off a debt, was reasonably common in Hungary (alongside the usual hereditary holdings, and office holding that you held while you were in an office appointed by the king). And before you ask, yes, some powerful nobles used it to permanently acquire a castle from a weaker peer.