I pwned those machines left and right growing up. I'd go somewhere with the parents and come back with a few stuffed animals for a few quarters worth of play if I saw a machine.
What's the secret? Simple fact is, not everything in there is grabbable. What's easier to pickup - a box filled loosely with stuffed animals, or one jammed with animals by hand? Obviously, the former.
Just look for animals not stuffed in - sometimes you got lucky and the employee had just thrown a bunch in without stuffing it - pretty easy pickings. Otherwise, you'd have to find the "loose" one. What was even easier was waiting behind someone else to play it. I'd almost always see someone grab an animal, it falls out when the claw goes back up or moves, and they leave. That's pretty much a gimme since you know it's loose.
Another thing is lining up the claw. Imagine the front of the machine where you are controlling the joystick - moving the claw left and right is the X-axis, moving it forward (away) or back (towards you) is the Y-axis. Most people would line up the X-axis and eyeball the Y. I'd line up the X, line up the Y, then move around to the side of the machine and make sure the Y-axis was lined up too. (I can say most people are too close to the front of the machine, at least from my own experience in making adjustments to the Y)
The places like Chuck E Cheese, Dave and Busters, or specific family oriented joints tended to have the most tightly packed IMO just b/c they knew kids were running free reign with assloads of tokens. The best pickings were the quasi arcades like in movie theaters, where teens making minimum wage just threw a bunch in.
With regards to weight - that does make a diff in some cases if it was heavily weighted on one end of the animal. IE, a large head or body and the claw only grabbed the lighter part. If the animal is laying flat, you had a better chance of keeping it in the claw once it raised up.
Those were the good days. I took my 7 year old daughter to D&B a month ago...still got it

She was amazed I snagged three things for her.