QuickTime Player > Export Clips

If you're OK with recording your mic audio and system audio pre-mixed, you could download the free utilities Soundflower and LineIn. Soundflower appears as both an input and output device, and echoes whatever it hears when you send it output to its input. This means that you can set your system output to Soundflower, as well as use LineIn to echo audio from your microphone to Soundflower, and use Soundflower as the audio input to your Quicktime recording.

There is one disadvantage to doing this though: because of latency, you don't want to hear yourself on a short delay as you're talking, which means that since you're piping both your computer's audio and your own audio into the same Soundflower device, you're not going to want to monitor it. Which, in turn, means you won't be able to hear the sounds your computer's making (but they will be recorded).

If you don't want them premixed, you can use the same setup minus the LineIn to record system audio on the QuickTime screen recording, and use a separate app like Audacity to record the audio from your microphone. You'll have to start the recordings separately and align them by hand in iMovie or whatever you're using to edit. If you're doing this and if you want to hear your system audio, open up SoundflowerBed and from the menu bar drop down select the audio device you want to send the Soundflower audio to.

I ended up just opening each file in QuickTime and exporting each clip manually. It wasn't the optimal solution, but did get the job done.

For future viewers, showing package contents on the video file did yield a set of individual movie files in MTS format. I defaulted to manually exporting each clip, however, because I wanted the files in MP4 and didn't have access to a video converter app. However, that would have been a viable option as well.