It doesn't give any unit two turns in a single day -- it just changes when in the day the units get their turn.
For instance, the turns for the Transylvito (T), Plaid (P), and Jetstone (J) sides are in this sequence:
(N) - T - P - J - (N) - T - P - J (N) - T - P - J - . . .
(N) = night; T = Transylvito; P = Plaid; J = Jetstone
Until now, the units in the war had all been on or allied to the Plaid or Jetstone sides, so it looked like this:
(N) - P - J - (N) - P - J - (N) - P - J - (N) - P - J - . . .
By switching to Transylvito alliance at the end of their turn, the strike force adds Transylvito to the turn sequence in a way that gives them two turns in between two of Plaid's turns -- once:
(N) - P - J* - (N) - T* - P - J - (N) - T* - P - J - . . .
*strike force's turn
Note that if the strike force switches back to Jetstone alliance, the extra move is "given back", with Plaid getting two moves between two strike force moves:
(N) - T* - P - J - (N) - P - J* - (N) - P - J* - . . .
*strike force's turn
If all goes according to plan, that will be a moot point, since it will be after the strike force dances on His Toolishness' little square head. If not, it might be a bit of a complication....