What is the 'Downstream Clearance?'
The definition you quote is pretty clear to me. But let me try to explain some of the background to help you understand it.

An ATC clearance always contains a clearance limit, which is the point to which the flight is cleared. Before reaching the clearance limit, the flight must obtain a clearance for the next portion of the flight. If that is not possible, the flight must not pass the clearance limit.

In most circumstances, a flight will receive its ATC clearance on ground before departure, and the clearance limit will be the destination aerodrome. The flight is then allowed to depart and continue all the way to its destination. This can be done because there is reasonable assurance that ATC will be able to coordinate the flight all the way through to the destination as the flight progresses.

In some cases, a clearance for (a portion of) a flight cannot be issued very far in advance. One example is clearance to cross larger portions of ocean (oceanic clearances), which are typically based on procedural separation and require the crew to provide very exact time estimates, which is impossible to do before the flight is airborne. Something similar is likely to be the case for certain military flights where some sort of diplomatic clearance is required to enter a countries airspace. Or maybe a flight has to cross the border between two countries that do not cooperate (a bit hypothetical in this day and age, but it might exist somewhere).

The definition of a downstream clearance is quoted in the answer. It means that, while a flight is flying in airspace A heading toward airspace B, the crew must obtain a clearance from the ATC unit in charge of airspace B before entering. This is because the flight will not be coordinated between ATC A and ATC B like most flights. How exactly this is done probably varies from place to place. The crew might contact ATC B on a secondary radio or CPDLC, or maybe ATC A can forward the request and relay the clearance from ATC B. In any case, the crew is responsible for not passing the valid clearance limit before a clearance to continue has been received.