Why should an area vector point normal to the surface? Why is it that the direction of an area vector should be always along the normal drawn to the surface? Can't it also be some other angles with the plane?
 A: This convention is extremely convenient when doing things that physicist often like or need to do, such as computing a flux through a surface. When the area vector is chosen normal to the surface, one can simply take use an dot product to get what you're looking for.
In the context of differential forms, this also turns out to be the natural definition, since - at least in 3-space - the surface vector is essentially a cross product of two vectors spanning the surface.
A: In this context, this is just a natural definition that turns out to be useful. For example, the flow rate of a fluid through a plane is the dot product of the fluid velocity with the area vector.
[There is a more mathematically sophisticated way of understanding this, which is that the vector area is really a bivector or two-form. In three-dimensional space, this is equivalent to a vector, but not in more general situations]
A: It's actually the "best" way to define a plane (or a hyperplane), that is, defining the plane by where it is not! Think about it: if a plane stretches across the floor, then it's not going up to the ceiling. So the direction from the floor to the ceiling is a unique direction that the plane does not exist in. Since conventions are about convenience, this is the most convenient way of doing it (for most applications, in pure and applied math/physics). 

Can't it also be some other angles with the plane?

You can define the plane with vectors other than the normal, but it will be much more complicated than using the vector that points in the direction where the plane does not extend into (this would involve cross-products as other answers have stated). This definition of a plane in 3D space is algebraic, but there exist (and these of course predated the algebraic notions) geometric definitions as well which might make more intuitive sense. 
