Since, ABCDEF is a regular hexagon, all the sides are equal. The lines joining the center to the vertices OA, OB, OC, .. OF divide the angle at O in equal proportions. Hence angle 360 deg. is divided in to 6 equal parts.
Hence AOB, AOF, BOC, COD, DOE, EOF each = 360/6 = 60 deg
In the triangle OAB, OA = OB, as they are equal to radius of circle. Hence OAB is an isosceles triangle. Hence OBA = OAB .
OAB = (180 - AOB )/2 = 60 deg
Hence all angles in OAB are equal. Hence it is an equilatera l triangle. Hence AB = radius of circle.
In other triangles also, you can prove in a similar way, OBC= OCB and BC = radius. and so on. Hence, all sides are equal to the radius of the circle.