Here's the so-über-short-it's-almost-misleading version:
Meek STV does calculations in rounds (or "iteratively," for you programmer types). In the first round, all votes count for the candidate marked as the first choice. The system figures out how many votes are needed to win. If anyone gets that many votes, he wins, and any "extra" votes he got are handed out to the voters' second choices. If nobody wins, the weakest candidate gets cut and all votes he received go to the voters' second choices. Then the next round starts. Rounds keep going until enough people have been elected.
Or, if you're into explanatory videos, check out Politics in the Animal Kingdom: Single Transferable Vote on YouTube.
As you might guess from the name "Single Transferable Vote," every voter gets one vote. But that doesn't make sense; you get to vote for three people in the election, right? Don't be fooled! In Meek STV, every voter's one vote might be split up into fractional bits and divided amongst the candidates or even thrown away. More on this later.
As with any voting system, there exists some threshold for victory, and any candidate who reaches the threshold is considered "elected." Meek STV calculates this threshold ahead of time. If a candidate gets more than enough votes to be elected, the difference between the votes received and the threshold is the "surplus." The candidate keeps just enough of everyone's vote to stay above the threshold; the rest is given to the voters' next-most-preferred choices in "redistribution."
Example time. Let's say that 100 voters select candidate A as their first choice. Let's also say that the threshold is 25. At the end of the first round, candidate A is considered elected. Since the threshold is 25, candidate A only keeps 25/100, or a quarter, of the votes he got. But this doesn't mean that 75 of the people who voted for him transfer their votes to their second choice!
Instead, candidate A keeps one fourth of every one of those hundred votes he got. Then, all 100 voters get to transfer the remaining three quarters of their votes to their second-choice candidates.
In short: when someone you vote for gets elected, you lose a fractional bit of your vote based on the threshold and the total number of votes the candidate got.
Okay. But what if there are no surpluses to redistribute at the end of a given round? In that case, whoever has the lowest number of votes — let's call him candidate Z — gets thrown out of the election as if he had never participated in the first place. Whatever votes Z did accumulate get redistributed to the voters' next-most-preferred candidate, and they're worth just as much as they were when they were assigned to Z.
In short: when nobody gets elected and your vote is counting for the weakest candidate, your vote gets transferred to the next candidate on your priority list.
Finally, what happens when a vote is already on a voter's third choice, and it's time for redistribution? The remaining portion of that vote just gets thrown away, not counted towards any candidate. Any slivers of votes that get thrown away are still useful for one thing: they count towards "excess" in the threshold calculation.
In short: when everyone you voted for has either been elected or eliminated, any fraction of your vote that hasn't been used yet gets thrown away.
Note that, after any given redistribution, an already-elected candidate may exceed the threshold again; the algorithm re-redistributes votes to account for this.
Warning: optional math ahead that explains where your vote goes
The ratio of the threshold to the total number of votes a candidate gets is called the candidate's weight, or w. In the example above, wa is 25/100, or 1/4.
Say that your first choice was candidate A, your second choice was candidate B and your third choice was candidate C. Here's where your vote goes, assuming nobody you vote for gets eliminated:
B: (1 - wa) wb
C: (1 - wa) (1 - wb) wc
excess: (1 - wa) (1 - wb) (1 - wc)
What if B got eliminated? Here's the adjusted breakdown:
C: (1 - wa) wc
excess: (1 - wa) (1 - wc)
(This concludes the math.)
Alright, so how is that threshold calculated, anyways? It's a lot harder than it is in older STV methods; actually, it's the reason the algorithm requires a computer. Meek STV is the only method to change quota mid-process. The quota generated at the start of every round by this expression:
total number of voters - excess
available seats + 1
Note that total number of voters and available seats are constant, so the threshold is only affected by excess once tabulation begins.
Partial list of sources:
Wikipedia on Wright STV
Wikipedia on Meek STV
An article from The Computer Journal describing Meek STV (PDF)