The object for which you need to find the electric field is a uniformly charged sphere. Uniformly charged means that at every point on the sphere the charge density is same.
Suppose someone blind-folded you and then he rotated the sphere in some arbitrary fashion about the origin(assuming your sphere has origin as the center). Then he takes off the blind fold.
The question is: is there any way you can determine if he rotated the sphere or not by measuring its charge density at each point or some geometrical property of the sphere?
The answer is no.
Thus as the electric field is determined by the charge density ( this includes information about both the charge at every point and the geometrical distribution of charge) in space and this charge density has spherical symmetry, do you think it is intuitively reasonable for the electric field to have this symmetry too?
Now consider a spherical Gaussian surface around the sphere centered at the origin. Consider two different points on the Gaussian surface. If the electric field at these two points are different then this asymmetry should come from the distribution. But as the charge distribution has no such asymmetry they should be the same.
About the direction of the field:
Assume the field has some direction that is at some angle with the radial direction. Remember my example where I showed that any arbitrary rotation should be undetectable.
Draw a triangle on the sphere. Start from one of the vertices of the triangle and rotate the sphere along the triangle such that the current position of some point on the triangle should be the position of the vertex at $t=0$ at every instant of rotation. Make one complete rotation along the triangle.
The constraint that the rotation should be undetectable will give you the direction of Electric Field at every point on the triangle. When you complete one rotation and come back to the vertex you started with you will see that the direction you assigned earlier is incompatible with the direction you gave initially. This will be so unless the direction was radially outward.
Thus you cannot assign a unique direction for electric field other than the radial one which is compatible with spherical symmetry.
In the above picture you start from the top vertex. Observe two different directions for the field after one rotation.
I am sorry if I was a bit clumsy at explaining it or if I was really confusing.