How and why does momentum get transferred? Let us imagine an isolated system and there are two objects of whatever masses and size. Let one object move towards the other object with constant velocity. We know, that when they collide, the moving object applies impulse on the other object and the other object starts to move. We know this because there has been a change in momentum in the moving object , because some or full momentum is transferred to the second object. But this leaves me with a question.
Q- I thought that something can move another object only when it applies force. Force is applied only when there is acceleration. Acceleration happens when there is change in velocity. I see that velocity is changed in the first object 'cause it has collided and transferred some of the motion (momentum) to the second object. 
But why and how is momentum even transferred?
 A: As you can see in my comment to your question, momentum transfer is due to electric repulsion. What do you mean by "how does repulsion transfer momentum?" Repulsion is a force, and force is equal to change in momentum, it is defined as $\vec F=d\vec p/dt$.
Consider the simple case of two cubes colliding face to face, when one of them touches the other with velocty $\vec v$, both objects feel a force of repulsion (due to their electrons interacting via the electric force), the stationary object feels it in the direction of $\vec v$, while the other feels a repulsion in the opposite direction, but with same magnitude. This clearly makes the moving object slow down while making the stationary object speed up. 
That is the transfer of momentum via contact forces, which you can see and check everyday, and it happens because of Newton's second law of motion: $$\vec F= \frac {d\vec p}{dt}$$
Because of the electric force: $$\vec F=\frac {k \ Q \ q}{r^2}\hat r$$ 
And because of the fact that matter is made of atoms with a negative electron cloud around them, making the outter surface of objects reppel.
A: But why and how is momentum even transferred? That is a good question. Due to transfer of momentum, stuff move right? So we can imagine momentum as the energy possesed by moving objetcs. Well, that is not the textbook definition but it would be better to imagine it as that.
So, when a moving object strikes a stationary object, why was energy transfered? The truth is, energy wasnt transfered. A better term would be, energy was induced.
Let me explain. When the moving object strikes the stationary one, the electrons come really close to one another. Then they repel one another. The electrons in the stationary object repel the electrons in the moving one. The force of repulsion is directly proportional to the speed of the moving object. Why? The faster the object, the closer the electrons will be pressed together when it strikes the stationary one. So, the repulsion force will have to increase to compensate the exclusion principle. This repulsion force is just enough to lower the velocity of the first object to levels acceptable with the principle. So, the first object either slows down, or stops. In our terms, it loses its momentum. 
  But we remember from Newtons third law. Force always acts in both ways. The electrons of the stable object push back against the moving object. But, this force also acts in the opposite direction in which the electrons are applying force, in other words, the direction the moving object was moving in before impact. This energy gained in this manner by the second object, is equal to the energy that was negated by the pushing electrons from the initial moving object. We can say, that energy in the first object was transfered to the second one. There we have it. Momentum has transfered, we think. But, thats not what really happened. But its similar, and weird at the same time. I hope i answered your question.
