On the night of the election, well over 100,000 non-overseas absentee ballots in Florida were counted. Traditionally, absentee ballots favor Republicans, and yet Gore only trailed Bush by a few hundred votes after machine counting.
Therefore, the only way that Gore could be trailing Bush by only a few hundred votes given the fact that absentee ballots favor the Republican candidate is if on Election Day, more people actually went to the polls and voted for Gore. This is to be expected, since Gore's polling results went up greatly in Florida in only the last few weeks before the election.
So, why is manual counting important? Because, while machine counting of punch ballots is a fine way of determining an accurate percentage breakdown on votes, it is inaccurate in determining the actual number of the votes! It is accepted knowledge that machine counting of punch ballots has a 5% failure rate... which means that 1 person in 20 is disenfranchised from their vote. This can be very important when it comes to tallying close elections, especially since punch ballots were primarily used in large metropolitan areas of Florida that tend to swing Democratic. Compare this to the advantage that Bush would have when in six smaller yet highly Republican counties in Florida, they counted all the votes by hand during the recount process. In those counties, everyone's vote counted, while in the other metropolitan counties, only 95% of the votes counted.
Lets say that on election night in Florida, about 3,000,000 people used punch ballots that have not as yet been manually recounted. (A guesstimate... there were different voting methods in different areas, including "fill in the blank" ballots that are considerably more accurate than punch ballots.) That would mean that there were about 150,000 ballots uncounted by machine counting. If we were to VERY conservatively assume that 48% of the ballots in these largely metropolitan areas were for Gore and that 47% of the ballots were for Bush, that would give Gore an advantage of 1500 votes, which would clearly be enough to win the election. (I suspect that I am underestimating this total by at least 1500 additional votes.) Statistically, since other voting methods are considerably more accurate, it is very likely that a full scale count of all punch ballots in Florida (or even an entire recount of the State of Florida) would also show that Gore won the election.
That is why the Republicans are fighting this tooth and nail. They will win the election if they can just figure out a legal way to deny those 5% of punch ballot voters their vote. Right now, it looks like they just might pull it off...