Why does friction remain constant with larger surface area?

[takando12]

My books and my teachers tell me that the frictional force is independent of the area of contact. I am also aware of the equation F=μN and that area has no part in it. But it's just so confusing and counter-intuitive.
Friction is because of the the interlocking of the irregularities of two surfaces (At least that's what i think it is). So doesn't more area mean , more interlocking and hence more friction?
Please do help me understand this.

 

[tommyxu3]

For an object whose mass is constant and shape can be deformed, the larger the one of the object's surfaces is, the less the gravity that unit area carries gets, so the frictional force remains also remains constant.