Reeeealy??
Fat is caused by consuming more energy than you burn
Its as simple as that.
So yes if you give your dogs lots of treats and dont reduce their meals or exercise them more then your dog will gain weight.
But many pet owners give their dog some kind of treat, I know lots of people who will give their dog a dentistick or boneo or something in the evening to keep them occupied. This is WAY more than my dogs get for training, I give my dogs lots of treats when training that is true - but - they are tiny treats. For example something the size of a cocktail sausage gives my dogs up to about 25 treats or I use tiny slivers of dried liver 1/2 the size of my pinky nail.
Mental exercise uses energy as well - so learning things IS exercise
Also if you train your dog to walk nicely with you or be happy in different situations then you are able to take them more places and walk them for longer walks so they are less likely to get overweight.
Myth busted - overfeeding/under exercising dogs makes them fat