It's geometry and you can get a petty good intuition by building simple models of the molecules
Saturated carbon-carbon bonds have two key properties of relevance. The angle between a pair of bonds on a carbon prefer to point to the corners of a perfect tetrahedron (~109.5 ° between each bond) if there are no spatial or geometric constraints. Also, C-C bonds can rotate freely unless something else in the molecule prevents that rotation.
In small cyclic aliphatic molecules (cyclopropane to cyclopentane) there is no easy way to satisfy the tetrahedral preference. No combination of bond rotations can give C-C-C bond angles of 109.5 °. If you build skeletal models of the compounds with tetrahedral carbon atoms this is easy to see: cyclopropane is rigid and flat; cyclobutane can twist slightly away from planarity but can't get those bond angles close to the unstrained angle, cyclopropane has slightly more freedom but is still pretty rigid.
But, from cyclohexane up there are plenty of extra freedoms. Cyclohexane can twist into a "boat" or "chair" shape and those give essentially unstrained bond angles at the carbons (the angles would be 120° in a flat hexagon, but when the bonds can also twist this does not constrain anything as enough combinations of twists are available to give an unstrained result in contrast to any of the smaller cyclic compounds). Plus, the twists cost little energy so they can interconvert easily given the amount of energy available at room temperature. Again, this is easy to see with physical models.
Higher cyclic compounds have even more freedom because there are more bonds to twist to give the ideal bond angle at carbon. So the increasing flexibility is a product of the fact that bond rotation is easy to do in the absence of geometric constraints. The more bonds you have to twist, the larger the number of conformations of the molecule available to give unstrained angles.