Where does the following usage of worry come from?
One of the connotations of worry appears to be quite different from its more common usage : to be or cause to be anxious or uneasy, bother.
Worry:
(transitive) to touch or poke repeatedly and idly, touch or rub constantly as in
"The old man worried his beads"
[Vocabulary.com/dictionary]
The above sense appears to suggest a sense of relaxation rather than uneasiness.
I could not find any reference to the above usage in Etymonline.
Worry LEXICO (Oxford)
- with object (of a dog or other carnivorous animal) tear at, gnaw on, or drag around with the teeth.
‘I found my dog contentedly worrying a bone’
2.1(of a dog) chase and attack (livestock, especially sheep)
‘a farmer shot a dog that had been worrying sheep’
2.2worry atno object Pull at or fiddle with repeatedly.
‘he began to worry at the knot in the cord’
Etymology LEXICO (Oxford)
Old English wyrgan ‘strangle’, of West Germanic origin. In Middle English the original sense of the verb gave rise to the meaning ‘seize by the throat and tear’, later figuratively ‘harass’, whence ‘cause anxiety to’ (early 19th century, the date also of the noun).
In concern to "he worried his beads", he's exhibiting a sign of anxiety by fiddling with the beads, as fiddling with the beads would be an attempt to help himself relax.
EDIT If you're referring to how "worry" changed from "strangle" to meaning anxious, then according to Quword:
The oldest sense was obsolete in English after c. 1600; meaning "annoy, bother, vex," first recorded 1670s, developed from that of "harass by rough or severe treatment" (1550s), as of dogs or wolves attacking sheep. Meaning "to cause mental distress or trouble" is attested from 1822; intransitive sense of "to feel anxiety or mental trouble" is first recorded 1860.
However, I have not come across anything which states how it changed from "strangle" to "the feeling of anxiety".
To worry someone or something is when you don´t let them be. It´s like the beads have got a feeling of their own; thus would like to be at ease, but the man won´t give them even that chance as he was probably having a sweet sensation caresing them. We can say it´s a poetic sentence, it´s just to give the impression that the man´s hands were restless.