This is tricky even for a native speaker, as it depends on knowing how straight razors are used. It used to be common to go to a barber to get a shave with a straight razor. Now most people use disposable razors, and so knowing how to sharpen a razor is no longer common knowledge.
As the final step of sharpening a razor, a strop is used to polish and straighten the edge:
Dr.K. 15:35, 4 November 2007 (UTC) / CC BY-SA
A strop is often leather, sometimes with a little abrasive on it. Since a razor must be very sharp to cut hair, it is usually stropped before every shave.
Suspenders may be elastic like your dictionary says, but in the time of the story, probably leather was more common. Also suspenders may not hold up only stockings, but also pants. For American English (all Twain stories will use American English) "suspenders" will mean something holding up pants, like these modern leather suspenders from Colonel Littleton:
The leather straps from suspenders like this could be made into a strop, and that's what the barber was doing.