In professional wrestling, a "botch" is a "spot" or other move that doesn't go as planned.
Professional wrestling isn't "scripted" in the way you think - it's not like the two competitors sit down and meticulously choreograph every single second of the match. There usually isn't time, and pro wrestling also is a dynamic endeavor in which the competitors react to the crowd. A hot match might get called to be longer, one that is boring the crowd might get ended quicker. The "victor" may very well be dependent on who the crowd cheers more for. Instructions on pacing, timing and so forth come from the booth and are relayed to the performers via the referee, whose real job is to use his earpiece to do just that.
And the word "professional" is in the sentence for a reason. These men train for a significant amount of time to learn not only how to land without hurting themselves, but cue each other without giving away the cues. Unless you're John Cena in which case you tend to talk loud enough that ambient microphones can pick up you talking to the other performer much of the time.
Things don't always go according to plan. Several bad things can happen, resulting in a botch.
You don't even need a move to go wrong. Unscripted monologues can go south fast, people mis-speak, say words that monagreen to others, or in the case of Booker T get so animated that they say on live TV, "HULK HOGAN, WE'RE COMIN FOR YOU, N*GGA" before realizing he's not supposed to use that kind of expression on live television, and instantly and visibly cringe while the others smile stiffly and try and distance themselves from what just happened. (The best comment on this so far was The Blue Meanie suggesting that Hulk Hogan was more... mahogany than anything else, given his serious and everpresent Floridian tan.)
Sometimes a move is poorly executed. Someone's off balance, tries a move they're not strong enough to do, or what have you, or tries something risky like jumping onto a rope at speed and simply slips, falls, drops someone on their head rather than upper back, etc.
At speed, sometimes someone mishears or misreads a cue and goes for one move and the other person responds with the other. Instead of jumping into a scoop slam, he falls over thinking it's a clothesline. Oh dear. This is one of the worst kind, because you've destroyed any ounce of credibility you've engendered.
Sometimes there's an equipment malfunction of some kind. Pro wrestlers like to put each other through tables, but sometimes a table proves insanely robust and the match then breaks the fourth wall as the wrestlers try to improvise spot after spot to try and smash through the table - because the unwritten rule is once a table goes up, someone goes through it. Occasionally a table collapses with someone on it, which can ruin a spot as well. Microphones can go dead, the wrong music plays, pyrotechnics go off at the wrong time. Owen Hart was killed when a flying rig gave way and dropped him several stories onto a ring post. Hitting the rope and turnbuckle, it snapped his body back and tore his aorta.
Sometimes human tissue gives way. One famous botch was caused by Vince McMahon running into the ring and tearing a quadriceps muscle in doing so, leading to him sliding into the ring and staying down - delivering his monologue in a seated position while maintaining one hell of a poker face with the agony of a torn leg muscle making it literally impossible for him to stand. To preserve his badass character, he chose to try to walk out under his own power and tore the muscles of the other leg. Triple H has torn a quadriceps muscle, and The Undertaker infamously wrestled for over half a match with a broken ankle. And there's horrifying footage of Sid Vicious snapping the bones of his lower leg in half and lying there with his foot and half a calf flopping around.
And sometimes there's just some ambient condition. People throw anything from lit cigarettes into the ring (trying to land them on the canvas just before someone's dropped there) to water cups. People get hurt, or people get very hurt. Droz is a permanent quadriplegic because he was wrestling someone who was trying to execute a "bulldog". He got Droz in a side headlock and then ran the length of the ring. Slipping on water, the guy holding his head fell straight down and Droz's neck broke, causing spinal cord damage.
Botches usually just make someone look rather silly. In the old days people just catcalled "you fucked up!" in a repeated chant until the crowd tired of it or were brought back into the affair with an awesome move. But with television, and then the advent of the Internet a humiliation can last for a very very long time. Almost everyone has heard of the time Fred Ottman was brought in as "The Shockmaster" and they decided to introduce him by having him smash through the wall of a set like "Captain Kool-Aid". Except that the guys that built the set didn't know Ottman was meant to do that, so they put a piece of 2x4 at around shin height within the wall to keep the wall square. Net result was instead of dramatically smashing through the wall he tripped up through it, landed on his face and the helmet he was wearing rolled off and away. Ottman tried to keep his composure, replacing the helmet and standing up - but that just emphasized that the whole gimmick was this huge fat guy wearing a Star Wars Halloween Costume Stormtrooper helmet spray painted with silver glitter paint.
And thanks to the internet we have "Botchamania", at this time of writing a compilation of spots gone wrong that's almost 400+ episodes long. Done by YouTube poster "Maffew", the compilation has given way to several memes (the most famous being Jim Cornette looking pained with the subtitle "fuck this company") and Maffew is a minor celebrity who's received shoutouts from indie and some mid-card pro wrestlers in cameos on Botchamania. They're usually set without comment to chiptune songs from 1980s Nintendo cartridges and suchlike.
Because of this, so long as nobody gets hurt, botches are generally seen as something to laugh off and laugh with the audience when it happens. But that can be taken too far, as some of the horrible car wrecks of botched spot after botched spot on CZW can attest.