Can you export only the audio from a video?
Yes. Video by Zaweba exports sound only as MP3, WAV, AIFF, AAC or FLAC, so you can pull the audio out of a video or save a finished mix without the picture.

Yes, you can. Video by Zaweba exports sound on its own as MP3, WAV, AIFF, AAC or FLAC, straight from its export window. Put the video on the timeline, press Ctrl+E, choose one of the sound formats and export. The picture is left behind and you get a sound file of the timeline's mix, made offline on your own PC.
There are two quite different jobs hiding in this question. One is pulling the soundtrack out of a single video, for a podcast, a voice note or a piece of music. The other is saving the finished sound of an edited project, with music, voices and effects all mixed together. Video by Zaweba does both the same way, because what it exports is whatever the timeline plays.
Choosing a format
MP3 is the one to use when the file is going to be listened to, shared or uploaded somewhere that expects small files. It is compressed, so some detail is thrown away, but for speech and casual listening that rarely matters.
AAC is another compressed format, common on phones and in Apple's software. It is a good alternative to MP3 for the same jobs.
WAV and AIFF are uncompressed. Files are large, but nothing is lost, which makes them the right choice when the sound is going on to another program for more work, such as a music app or a podcast editor. WAV is the usual choice on Windows; AIFF is common on Macs.
FLAC is lossless but compressed, so it keeps every detail like WAV while taking less space. It is a good choice for archiving music or recordings you care about.
Exporting the sound
Import the video and drop it on the timeline. If you only want part of it, split it with S and delete the rest, or set the export range, since each timeline keeps its own.
If you want only the sound and nothing else, that is all you need. If you are tidying it up, you can do that first: Edit > Audio gain (G) sets or normalises the level, fades can be dragged in at each end from the small square in a sound clip's corner, and noise reduction and voice EQ clean up speech.
Press Ctrl+E, choose MP3, WAV, AIFF, AAC or FLAC, and export. In Export > More options you can set the sound channels (following the project, mono, stereo or 5.1) and the sample rate, and fill in title, artist and copyright metadata so the file is labelled properly in music players.
Things to watch for
The export follows the mix. Muted tracks are left out, and if music is ducking under dialogue, the exported sound ducks too. That is usually what you want, but check before exporting if you meant to keep only one part.
Unlink when you want just one sound. If you need the voice without the music, or the music without the voice, mute the tracks you don't want, or unlink the clip's picture and sound with Ctrl+Shift+L and delete what you don't need.
Loudness is levelled on export. Video by Zaweba measures and normalises the mix to its loudness target as it exports, which means a podcast or voice note comes out at a consistent level.
Several sound streams come in separately. When a video file holds more than one sound stream, each is placed on its own linked sound track, so you can export just the commentary from a recording with a second language track, for example.
Why do it in a video editor
A dedicated converter can strip the audio from a file too. Doing it in Video by Zaweba means you can trim, clean, fade and level the sound before it goes out, and the same project can then export the finished video as well. It is free software, offline and without an account, so your recordings never leave your PC.