Do you have Root Explorer? You could open the zip file and extract the swype.apk, move it into /system/app. Not sure if you'll have to click on the apk in order to install it, possibly.
I assume the zip file downloaded correctly and it contains the file. Might have been a dud download.
I think you're right Neets, that extracting the file should work for side-loading 3rd party apps, but I have a feeling that Swype is a 'system' app, so needs ADB to install successfully. Which is why, I think, that Doc supplies it as a signed zip file.
I'd be inclined to remove any traces of Swype, including any Titanium backup of it, and attempt to use the CWM recovery option to install it. I had to resort to Doc's zip to get it installed on my SGS because it refused to install any other way. But I have to add that I did a factory re-set before I was able to get much of anything to stick. BTW, 'My Files' is capable of installing .apk files.
I'd recommend the re-set and CWM approach as the one with the highest probability for success.
I'm on a vanilla Froyo install now, and while I'm sure it's not as fast or as smooth as the 7.5 on ext4 + OCLF, I think it'll make my update to 2.3 a little less painlful. But I won't be surprised to find out that I'm wrong. It's a pleasant feeling though when Kies sees the SGS and says, hey, have I got an update for you... and a backup and restore later, ta-daaaahhhh...
Roll on Gingerbread...
Edit: Events have overtaken the above in large part, so...