Unlike what is often assumed the in the realms of theoretical macroeconomics there is not always an infinitely deep market with constant buying and selling. In fact, one of the reasons Investment banks are so important is in the role they play in making markets for buyers and sellers of stocks and financial products of many kinds.
Lets think about liquidity in the current context - the liquidity of a, say, firm's shares is determined by the available quantity, indications of interest &C. We could construct a working difenition of liquidity as the difference between bid price and selling price (in the context of a firm's shares although the intuition holds).
In practice, the liquidity guarantee means that an entity involved in the exchange acts as the guarantor for there being bid prices and selling prices, a guaranteed quantity of shares available, and especially ensuring that the discrepancy does not become too great between bid price and selling price. This is of importance for, say, investors, since a smaller spread significantly affects the transaction costs, risks etc.
It is essentially someone (post-financial crisis it has often been the government) guaranteeing that a market will be made (with buying and selling prices in good faith) for a given time period (or indefinitely). It is a form of prevention of stocks becoming illiquid, almost like a form of insurance.
The application of a liquidity guarantee is far from limited to buying/selling shares in a firm of course (although I believe the intuition is the same) - a notable example has been the US government's temporary liquidity guarantee programme which was constructed in the aftermath of the 2008 financial crisis with the purpose of getting banks to begin lending to each other again - the purpose of the programme was "to decrease the cost of bank funding so that bank lending to consumers and businesses will normalize." and "to strengthen confidence and encourage liquidity in the banking system by guaranteeing newly issued senior unsecured debt of banks, thrifts, and certain holding company, and by providing full coverage of non-interest bearing deposit transaction accounts, regardless of dollar amount."
Banks who made out liquidity guarantees are essentially insurance salesman and hence are exposed to the downside if things catastrophize. Often this insurance money was being kept in markets that would also be affected by market forces that made the liquidity guarantee they were insuring catastrophize in the first place! (These problems were exacerbated by many other factors including very high effective leverage rates).