If a player doesn't have enough cash on hand to meet a rent, or other obligation, the person can try to liquidate his holdings to raise sufficient cash:
1) Sell houses and hotels back to the bank for half price.
2) Mortgage properties to the bank for half price.
3) Liquidate other assets such as "get out of jail free" cards (if house rules allow it).
If after taking these steps, a player still can't meet his obligation, then he has to turn over his (mortgaged) property to the landowner or bank and leave the game.
It is therefore implied, although not stated, that you can sell unmortgaged property at, say, full value, either to the landlord, or to a third party.
Some landlords will agree to accept property at full (list) value, or at another agreed value toward the payment of rent, so that they get unmortgaged property. It's perfectly ok for a landlord to say, "You owe me $1,000, but if you give me $400 plus Boardwalk (List price, $400), we'll call it quits."
Otherwise, there might be a competition between the landlord and third parties to buy property from the renter at prices to be determined. In essence, a bankrupt player can put his property at auction to try to pay his bills, as one might do in real life. And the "strategic" value of the property might cause it to command more than list price. Even the idea of keeping a weakened player in the game might be a good strategy for some people.
The only caveat is that if the "auction" process doesn't raise enough money to pay the debt, the trades will have to be "undone," and the debtor has to turn over his "mortgaged" property to the landowner. (In a real bankruptcy proceeding, a trustee controls sales.) The "scorched earth" impact of the debtor's mortgaging all property gives the landlord (or third parties) an incentive to work out "deals."
"Selling" in this context is different from buying. You couldn't buy (build) a house on a property that someone has just landed on to jack up the rent in "real life;" the house has to have been built beforehand. Therefore you can't do so in Monopoly. But in "real life," a person that owes you rent could sell his own (existing) house(s) to meet this obligation.
And about the part of the third player overpaying on a property to keep the debtor in the game. That's likely to benefit the landlord. Let's say that the rent was $1,000 and the debtor could pay only $500 after mortgaging all his property. If the third party buys, say (mortgaged) Oriental Ave. (list value $100, mortgaged value $50) for $500, the landlord actually gets the full $1000 instead of just $500, with the difference coming out of the pocket of the third party. Of course, the third party may have a "strategic" reason to pay $500 for Oriental, either to complete or prevent a Monopoly, but that's just part of the game.