There is a simple explanation in the introduction:
In short, the theorem states that no rank-order voting system can be
designed that always satisfies these three "fairness" criteria:
1. If every voter prefers alternative X over alternative Y, then the
group prefers X over Y.
2. If every voter's preference between X and Y remains unchanged, then
the group's preference between X and Y will also remain unchanged
(even if voters' preferences between other pairs like X and Z, Y and
Z, or Z and W change).
3. There is no "dictator": no single voter possesses the power to
always determine the group's preference.
A "rank-order voting system" is one in which all voters order the possible choices, from most preferred to least preferred. The idea is that based on everyone's votes, and some set of rules on how to evaluate those votes, you can come up with a decision for the entire group that respects those preferences, in some way that is considered fair.
All of these requirements listed above seem like fairly simple requirements for a voting system to have; if there's unanimous agreement about the ordering of two of the choices, then the voting system should order them that way. If a voter changes preferences about one choice C (maybe changing between them being ranked above or below D, or above or below one of A or B), without changing their ordering between choices A and B, that shouldn't affect the outcome of the A/B ordering that the voting system gives you. And finally, there is not dictator; no single voter is able to always determine the group ordering unilaterally.
The first requirement seems pretty obvious; if everyone is unanimous that we should pick one option over another, then the group should pick that option.
The second is a little trickier; but it basically states that there shouldn't be able to be "spoiler"; someone whose ordering in people's preferences affects the ordering of to other choices. Think about, say, the 2000 election, with Bush and Gore and a couple of third party candidates; now, that used just single choices per voter rather than a ranking, but imagine it used a ranking. So let's say Bush and Gore are fairly close, but 51% of voters prefer Gore over Bush. Their preferences for other candidates should not affect the fact that Gore wins over Bush. If I rank candidates Nader > Gore > Bush, that should not change the Gore/Bush result compared to if I ranked the candidates Gore > Bush > Nader.
And the last is pretty trivially desirable; generally a democracy wants to democratically make a choice, in which everyone's vote has the same weight in affecting the final outcome.
The fact that these conditions are impossible to achieve together is fairly profound, and means that almost any voting system will have serious flaws under certain circumstances.
All of these requirements listed above seem like fairly simple requirements for a voting system to have; if there's unanimous agreement about the ordering of two of the choices, then the voting system should order them that way. If a voter changes preferences about one choice C (maybe changing between them being ranked above or below D, or above or below one of A or B), without changing their ordering between choices A and B, that shouldn't affect the outcome of the A/B ordering that the voting system gives you. And finally, there is not dictator; no single voter is able to always determine the group ordering unilaterally.
The first requirement seems pretty obvious; if everyone is unanimous that we should pick one option over another, then the group should pick that option.
The second is a little trickier; but it basically states that there shouldn't be able to be "spoiler"; someone whose ordering in people's preferences affects the ordering of to other choices. Think about, say, the 2000 election, with Bush and Gore and a couple of third party candidates; now, that used just single choices per voter rather than a ranking, but imagine it used a ranking. So let's say Bush and Gore are fairly close, but 51% of voters prefer Gore over Bush. Their preferences for other candidates should not affect the fact that Gore wins over Bush. If I rank candidates Nader > Gore > Bush, that should not change the Gore/Bush result compared to if I ranked the candidates Gore > Bush > Nader.
And the last is pretty trivially desirable; generally a democracy wants to democratically make a choice, in which everyone's vote has the same weight in affecting the final outcome.
The fact that these conditions are impossible to achieve together is fairly profound, and means that almost any voting system will have serious flaws under certain circumstances.