Here's some that I found with some googling. All open source and should run on any platform (I've never done any video editing myself, so I have no idea if these are any good)
This one is a webm format, so it's actually a video. You can use software like miro http://www.mirovideoconverter.com/ to convert your recording to webm format. although alot of sites now will convert it automatically. gfycat's limit for video is 300m's on upload. You can also use open source video converters like handbrake to adjust the quality a bit. and Vlmc http://www.videolan.org/vlmc/ to crop edit your footage (from the same team as VLC Player)
A while back, I stumbled across VLMC - VideoLAN Movie Creator while looking for some basic video editing software. It had potential when I tried it, but does not look like it's been updated at all in quite a while. If you use it, save frequently because from what I recall, it was not very stable. For what it's worth, I ended up using Avidemux since it was basically trimming and splicing a bunch of YouTube videos.
I wouldn't say it's on the same level as they are, but it's certainly a very good alternative for those who wanting a kind of consumer level software without spending the money on the big boys. VLMC is also a great alternative for those who want something easy to use for basic editing on Windows, but again if you want the real professional level effects, plugins, and features I'm not sure anything can compare to Vegas and Adobe PP...no matter what OS you are running.
Perhaps VideoLAN Movie Creator (VLMC)? I stumbled across this a few years ago and tried it. It was a little unstable, so save frequently. But overall it seemed easy to use for creating linear videos. It looks like they have removed the downloads, but you may still be able to find them somewhere.
Don't listen to the other guy.
VLC have a video editor, VLMC. It looks pretty simple to use, which makes sense considering how well VLC works.
You can also use blender (http://www.blender.org/) for editing videos. Here's a tutorial list.
just download any video editing software (try http://www.videolan.org/vlmc/), load the film in as a layer. When the second song plays, drop the sound out for that layer. Load in whatever song you'd like to replace it with. Export as video.
Done.
Check out Open Movie Editor or AVS Video Editor. I've not tested them myself, so I sadly can't vouch for them.
There's also an open source project in the works by the VLC organization: http://www.videolan.org/vlmc/