The formula for moving mass in relativity is

M = M0 / sqrt(1 - (v/c)^2)

If v is much less than c, then we can approximate this as

M = M0 ( 1 + (1/2)(v/c)^2 )

or

Mc^2 = M0c^2 + (1/2) M0 v^2

This statement says - the total energy of a body Mc^2 consists of a constant part M0c^2 plus a correction term (1/2) M v^2, which is just the Newtonian kinetic energy. This shows why it goes as a square and also where the factor of 1/2 comes from. The next correction term would involve v^4/c^2, then v^6/c^4 etc.