The (naive) answer is, or rather used to be, simple: It tranlates to each sender having to wait a tiny fraction of a second. There is no way around this because that's how the network works at a physical level, but it is also not very much of a problem since we're talking about a few microseconds (micro, not milli).
The somewhat less naive answer would be that unless LAN also includes WLAN, your quote is outdated (wrong). On a WLAN, you still do not have many options. While one station is sending, no other station can send (unless they use a completely different non-overlapping channel, but that's really two networks, not one, so one would be cheating).
Computers in a modern "gaming" LAN will typically be connected via at least 100BASE-TX, more likely 1000BASE-T, both of which support full-duplex operation to begin with. This means you can send and receive on the same physical cable (well, a different wire within the cable, but still) at the same time. The notion of having to wait while the cable is busy is therefore not so clean-cut any more, since sending doesn't really interfere with receiving any more. Only the traffic coming from multiple other senders independently interferes.
Further, the cables will nowadays almost without exception be plugged into a switch (rather than a hub) which can, in principle (depending on the quality of the hardware) send and receive independently on every port at the same time, and which in principle (again, depending on the quality of the hardware) has enough internal bandwidth to process N times the maximum theoretical throughput for N ports. Your mileage may vary slightly since switch quality varies. Some very cheap switches will have N ports but only enough bandwidth corresponding to half as much data as can be pushed through N ports at the same time. Often, even that is (surprisingly) perfectly good enough.
Thus, you will still have some occasional, inevitable delays due to packets having to be queued when two computers in the network send at almost-the-same time to the same destination (the switch then makes that decision, not your computer's network card), but for the most part it will "just work". Note that even when it doesn't "just work", it works anyway, only just it takes a microsecond or two longer for the data to be received.