It comes from the fact that force is proportional to acceleration (F=ma). Then the factor of 1/2 comes from the formula for the area of a triangle, or equivalently, integrating a constant twice wrt time, gives term in t, then a term in t²/2.

Consider constant force acting on a mass initially at rest, for time t producing final velocity v and distance travelled x.

  F = ma                         [Newton's 2nd Law]
  v = at = (F/m)t, so mv = Ft    [momentum transfer is force times time]
  x = (1/2)vt                    [area under v-t curve: triangle formula]

  E = Fx = F(1/2)vt = (1/2)v(Ft) = (1/2)mv²

see also: https://tauday.com/tau-manifesto#table-quadratic_forms