If you apply a force through the “centre of mass” of an object that is free to move then the object will not rotate. It may translate but it will not rotate. If you apply a similar force to the object that is not directed through the “centre of mass” then the object, which before merely translated, will now rotate as well as translate.
Note the jargon word here is translate, which in this case means to move along a straight path from one place to another. Also “a force directed through the centre of mass” means the force is applied to the surface of the object and the direction of the force is pointing directly at the centre of mass. If the object is free to move the centre of mass will be the part of the object that moves in the direction the force is pointing.
The video shows you how to find the “balance point” of several objects. Sometimes this is the same as the “centre of mass”. In a gravitational field every part of the extended mass is attracted towards the centre of the Earth. So if you place your finger under the pivot point of a ruler the force due to gravity on one side of the ruler is equal to the force due to gravity on the other side and the ruler “balances” on your finger. The equal force situation is the same for the balancing bird at the end of the video. There is a difference between these two examples however. The centre of mass of the ruler is inside the ruler. The centre of mass of the bird is about 1 cm below the tip of the birds beak when it is balancing on your finger. So the balance point and centre of mass of the balancing bird are not in the same place.
Finding the balance point of the ruler, baseball and hockey stick utalises the frictional force between the surfaces. The frictional force depends not only on the characteristics of the two surfaces in contact but also on the normal force between them. Remember gravity is pulling all parts of the extended object towards the Earth so when one hand is further away from the balance point than the other it experiences a smaller normal force than the hand that is closer and is therefore more likely to overcome static friction and slip.
The demonstrator (me) says “weight” when perhaps the words “mass distribution” or “weight distribution” (mass distribution in a gravitational field) would be more appropriate but sometimes words get in the way of meaning and the words “weight” and “weighted” were used advisedly.