I noticed this strange action done by Verne at the ending of Back to the Future III, which appears to be a little indecent. This scene appear in duration 01:50:40 to 01:50:45
Was this a mistake or there is some plot explanation behind this too?
There was no plot explanation. There is much speculation (and I highly doubt this is going to show up in an interview or in DVD commentary since it should have been spotted by the Continuity Editor) that the actor needed to go to the bathroom. If you have kids in the 6-8 age range, which is approximately how old Dannel Evans was at the time, you know that kids are notorious for needing to go to the bathroom 10 minutes after being asked, as in when you're leaving for a lengthy car ride. Considering the pained look on his face and that he's mouthing something to someone while looking like he's about to cry (you need to watch the entire clip to see what exactly is going on), I'd say he's in desperate need of a bathroom break.
BTW, this has been floating around the Internet for almost 10 years. Most references find it amusing to call the kid "creepy", so that's just what stuck.