Why would you stop a recount? That reeks of corruption.
Why would you stop a selective recount aimed at finding more votes in a specific set of counties that tended to lean toward one candidate? Hmm...
The above is a one-sided interpretation, but it's not an unreasonable one. The situation was that Florida had voted with punchcard ballots, and there were problems with a percentage of the ballots, both "overvotes", cases where more than one candidate was punched, and "undervotes", cases where no candidate was punched in a race. But it turns out that when you're talked about punched cards, "punched" is an ambiguous term. There were cases where chads (the little rectangular pieces of paper that get punched out) were left barely attached ("hanging chads") and cases where they were partially punched but still fully attached ("dimpled chads"). And everywhere in between.
Gore's team called for the targeted recount, arguing the theory that the targeted counties had more elderly populations who were more likely to have failed to fully punch their choices, resulting in "undervotes" when the scanning machines read them, i.e. the machine saw no holes for a race and decided that the ballot didn't vote for anyone. They argued that human examination of those ballots could clearly see in some cases that a position was punched and that a recount should be done to count those. The Florida State Supreme Court ordered a selective recount of the counties the Gore team thought should be recounted, to correct correctable undervotes.
Bush's team argued to SCOTUS that the selected counties were all Democrat-leaning (they were) which meant that the voters in the Republican-leaning counties that had not been selected were not receiving equal treatment under the law. The initial statewide count was done by machine, but the recount which would be more "permissive", finding votes where the machine wouldn't, was only being done in counties where the newly-found votes were more likely to be for Gore.
SCOTUS stayed the recount while they decided how to handle this, then ruled a few days later that the recount was unequal treatment, that a proper recount would need to be statewide and it would need clear rules for how to count ambiguous ballots, rather than each county defining its own rules. But by then the clock had run out anyway. The court's rationale for staying the count while they decided was that if the count proceeded counties would release updates that would likely show the vote totals shifting, which would make rejecting the recount look really corrupt if it went against Bush and if they decided that's what should be done. "Oh, the conservative court let the count go forward until they saw it wasn't going their way, then they rejected it" was worse than "The conservative court stopped the recount before it produced any results", was the theory.
You can certainly argue that these published positions by the court weren't the real reason, but they're not without merit.
In any event, as a careful, methodical, independent recount determined months later, the Gore-requested recount would still have shown Bush won. An incomplete recount which resulted in no win would have been a Bush win, because the state legislature was voting to send Bush votes for that case. An incomplete recount in which Florida failed to submit a slate would have been a Bush win, because the US House would have picked him. The only scenario in which Gore won was a statewide recount that also tried to tally "overvotes" -- having humans try to discern which of multiple punched chads was "most" punched, but no one at the time thought that would favor Gore, and it wasn't even being discussed.
Aside: "Corruption" is the wrong word. Corruption specifically refers to bribery and other compensation-related schemes (not necessarily monetary). "Partisanship" is a better word.
