A sphere has an infinite number of symmetries, which can be divided up into two categories:
If a charge distribution has spherical symmetry, its electric field must have spherical symmetry as well. What would such an electric field look like? For one thing, the electric field must be radial: it either points outward from the center of the sphere, or inward. To prove this, suppose the electric field at some point outside the sphere wasn't radial, but pointed off to the side. If we rotated the sphere around the sphere's axis that passes through that point by 180 degrees, then the sphere would look exactly the same and the point would be in the same place, but the field would point in a different direction. This is a contradiction, so the field can't do that: the electric field at any point must lie along the rotational axis of the sphere which passes through that point, which means it points radially.
Another result of the spherical symmetry is that the electric field's magnitude can only depend on how far one is from the sphere's center; it can't depend on latitude or longitude, because once you find the field at one point, you can rotate the sphere and move that point to any other point which is the same distance from the sphere. So two things are true for any spherically symmetric charge distribution:
Suppose we smear charge out evenly on the surface of a sphere, creating a spherical shell of charge. This distribution has spherical symmetry so its field must be radial; therefore inside the shell the field could only look like one of these two pictures. However, if we apply Gauss' Law to these two figures, we see that both are impossible. If we draw a Gaussian sphere inside the spherical shell, as shown by a dashed line, then the total flux through that sphere is either positive (in the first picture), or negative (in the second). But because there is no charge inside the shell (it's all on the surface), the net flux through this Gaussian sphere should be zero. The only way we can have a spherically symmetric electric field with zero flux through this Gaussian sphere is if there is no electric field at all. Thus
This result says that the electric field inside a spherical shell is zero, even really close to the surface where the charges reside. The figure to the right shows how this works. We can classify each charge on the surface of the sphere as being to the left or the right of the target. The "left" charges create electric fields whose horizontal components point to the right, and the "right" charges create electric fields whose horizontal components point to the left. If we look at targets closer to the right side of the sphere, the field created by the "right" charges gets stronger because the target is closer to them. However, by moving the target to the right, there are now fewer "right" charges, and more "left" charges, than before, and we end up with a tug of war between "fewer but stronger" and "more but weaker". It is not always clear which side will win; for instance, if the charges are merely arranged in a (2D) circle, then "fewer but stronger" wins. However, Gauss' Law tells us that the sphere has a peculiar geometry so that these two sides perfectly balance each other, so that the electric field cancels out everywhere inside the sphere.