For example, moment of a force is represented as r × F. How is then the resultant in that direction ?
Imagine the two vectors as pointed sticks. There is only one plane in which both sticks are tangent to the plane (they do not stick out of the plane). This is the plane that is said to be formed by the two vectors. The cross product of those two vectors is perpendicular to that plane. For example if the two vectors are horizontal, then the cross product must be vertical (directly up or directly down); if the two vectors are in the $xz$ plane, then their cross product must be in the $+y$ or $-y$ direction.
In the case of moment or torque, the direction of the (net) torque vector is the axis around which the torque would cause something to rotate. (There's really no good way to define a direction of rotation other than by defining the axis.)