I caught this clip in the movie "Sun Valley Serenade" that features the Glenn Miller Orchestra playing "Chatanooga Choo Choo." It's an annoying song, mostly, but was quite popular in the day.
Besides, that, we get to see the guitar player up close. I think the guitar player is Jack Lathrop, anyway, you can really see his left hand lift off the strings with each strum. Most of the swing guitar players did this to get that short staccato sound.
To do that, just strum the chord (a chord with no open strings, mind you), and right after you strum, release the pressure of your left-hand fingers so that the strings lift off the fretboard. You should still be touching the strings, just not applying enough pressure to push them against the frets, thus the strings are muted.
For that swing or big band guitar sound, you do this "Four to the floor," in other words, on every quarter note.
The odd thing about this clip though is that he's lifting his fingers ridiculously far off the strings on each beat. Maybe he was just playing things up for the camera. If you try this technique, your fingers should barely look like they're moving at all, unless you have incredibly high action such that your strings are very high off the fretboard. But that's unlikely unless you're Freddie Green.
You can see all this at about 24 seconds into this clip of the movie on YouTube:
