Sadly, if there's one thing I've learned about used lawn equipment, it's given to you because someone else was completely fed up with fixing it. I just fixed my mower again, for the 20 time this year it feels like. In the long run it's normally better to just but a new one vs fixing it. Mowers seem to be built as cheaply as possible these days and don't last. Sorry I was no help, but having just fixed my mower again, I feel you pain.
While that is often the case--I have scored many free riding mowers that were left out in the weeds after a relatively minor issue happened,and the owners not being mechanically inclined at all.either just gave up on it and bought another one--or worse,tried fixing it themselves and did more damage than it had already..
I have dragged home dozens of riding mowers and most were running the same day,all most of them needed was a good carb cleaning and some maintenence they never got before--some parts were usually NG,like a gas tank,or the mower deck was rotted away,but many I got looked like new still and only needed something stupid like a gas filter...others were hopeless cases that had already blown up,or were about too--those got stripped for parts..
Many of the "newer" riding mowers have hydrostatic transmissions that die an early death when they lose some fluid from a minor leak and they stop propelling it,and the owner assumes its junk--sometime just a fluid dump & refill will get more life out of them--otherwise your better off just getting a parts tractor with a good one that had a blown engine and swap it in..
--also many of the later Briggs engines blow head gaskets easily and will smoke like a tire fire,and eventually let it run too low on oil and chuck a rod--if you can get it before it got run too low,its a fairly simple cheap fix to replace the head gasket--push rods,rocker arm studs,and valve seats often fail on the later Vanguard engines too...
In the case of the original poster,I believe Jeremy is correct--looks like you had an "automatic" (hydrostat) transmission and you were sold a manual transmission..most Craftsman tractors could come with either one,but to swap a manual into an automatic one will require a "donor" tractor that had a manual,to get all the different mounting brackets,a clutch pedal,linkages,the brake setup may be different--etc...its not a simple bolt in deal otherwise..
Also I have run into transaxles that may look almost identical,but had the drive pulley end up being higher or lower than the original ,that nixed any hopes of making it work,despite it bolting right up..others lined up OK,but the pulley was smaller than the original,and trying to cheat by using a shorter belt didn't pan out because the belt would grind against things in the way..