It can be embarrassing when your Apple Watch starts speaking when you don't want it to. One possible cause is the digital crown being snagged by clothing. The digital crown is what you'd call the winder on a mechanical watch and it rotates very freely and can easily catch on a sleeve cuff.
If you are experiencing unwanted VoiceOver speech from your Apple Watch check these settings in the Watch app on your iPhone:
Display & Brightness / Wake on crown rotation. You can easily wake your watch other ways - by tapping the screen or clicking the digital crown; consider turning this switch off if it's likely that clothing will catch on the crown.
Accessibility / VoiceOvert /Digital Crown Navigation - I suggest turning this off since if your watch wakes unintentionally and clothing catches the crown it will move VoiceOver focus and VoiceOver will probably speak. You can navigate using swipes on the screen.
Accessibility / VoiceOver / Speak on wake - you may find this handy but the watch will obviously speak whenever the watch is unintentionally woken. Turning off the first two settings may do the job but, if they don't consider turning this one off too.
Although it has nothing to do with catching on clothing, some people unintentionally wake their watches with a wrist raise. It's a useful feature if your other arm isn't free to operate the watch but it's another way you could unintentionally wake the watch. It can be turned off in the Display & Brightness section of the iPhone Watch app if you don't need it and it causes problems.
If you are somewhere where unwanted Apple Watch speech would be super embarrassing I suggest you temporarily instruct Siri to turn off VoiceOver as well as turning on Do Not Disturb.
A note for Apple Watch Ultra owners:
The Apple Watch Ultra models have a Siren feature designed to summon assistance from afar. Apple states that it can be heard from 600ft away and recommend avoiding starting it in an enclosed space. It is extremely loud and it could be very embarrassing to set this off unintentionally. On the other hand you might feel reassured to have the feature to use in an emergency but in many situations it would be better to call emergency services first with Emergency SOS which can be accessed with a long press on the Side button, the one under the digital crown. The siren can be sounded by pressing and holding the Action button, the button on the side opposite the digital crown. I had a call from someone who had unintentionally turned on the siren, probably this way. If you prefer to disable this feature open the Settings app on your Apple Watch Ultra and locate the Action button settings. Turn off "Hold to turn on" under Siren. There are other ways to turn on the siren but these are less likely to happen unintentionally.