How to create a spherical skybox???

That's just a sphere with inward looking faces. A normal sphere mesh (like the one that comes with Unity) has it's faces point outwards so when you view the sphere from the inside you wouldn't see anything. When the faces are reverted it's the opposite So from outside you see through the faces but from the inside you can see them. That's why you see the back of the sphere when you're outside the sphere.

edit

I quickly made a simple script that inverts all faces of a mesh at runtime. Just attach the script to a sphere GameObject. It can invert the faces as well as the normal vectors (so any lighting calculation would be correct). It also works on a cube or any other mesh.

 //MeshInverter.cs
 using UnityEngine;
 
 public class MeshInverter : MonoBehaviour
 {
     public bool InvertFaces = true;
     public bool InvertNormals = true;
     void Start ()
     {
         var mf = GetComponent<MeshFilter>();
         if (mf != null)
         {
             var m = Instantiate(mf.sharedMesh);
             Process(m);
             mf.sharedMesh = m;
         }
         var smr = GetComponent<SkinnedMeshRenderer>();
         if (smr != null)
         {
             var m = Instantiate(smr.sharedMesh);
             Process(m);
             smr.sharedMesh = m;
         }
     }
 
     private void Process(Mesh m)
     {
         int subMeshes = m.subMeshCount;
         for (int i = 0; i < subMeshes; i++)
         {
             if (InvertFaces)
             {
                 var type = m.GetTopology(i);
                 var indices = m.GetIndices(i);
                 if (type == MeshTopology.Quads)
                 {
                     for (int n = 0; n < indices.Length; n += 4)
                     {
                         int tmp = indices[n];
                         indices[n] = indices[n + 3];
                         indices[n + 3] = tmp;
                         tmp = indices[n + 1];
                         indices[n + 1] = indices[n + 2];
                         indices[n + 2] = tmp;
                     }
                 }
                 else if (type == MeshTopology.Triangles)
                 {
                     for (int n = 0; n < indices.Length; n += 3)
                     {
                         int tmp = indices[n];
                         indices[n] = indices[n + 1];
                         indices[n + 1] = tmp;
                     }
                 }
                 m.SetIndices(indices, type, i);
             }
         }
         if (InvertNormals)
         {
             var normals = m.normals;
             for (int n = 0; n < normals.Length; n++)
                 normals[n] = -normals[n];
             m.normals = normals;
         }
     }
 }