Because they "harvest" the organ when the patient is alive according to halacha. Brain death is not death. So if one donates organs it requires someone pulling a plug, which is the same pikuach nefesh for the one that the organs are coming from.
This.
I wonder why it took so long for someone to actually post the correct answer.
How in the world did this thread become so long when CLEAR CONCISE AND ACCURATE ANSWERS were given by
@hvaces42 to the main question and by
@aygart to many of the ancillary questions?