The first way shown way to complete this level is to get the bomb close to the pigeon.
To do this you need to only make moves in the area you want the bomb to be.
Use the bomb blow the metal cage away so that the pigeon is only in a wooden cage which can be destroyed by the built up rocket.
The bomb will not always cooperate and get to where you need it, but there is another method you can use (shown in the second play-through in the video)
Use the bomb to help build up the rocket booster more quickly (3 bombs blasts usually earn one booster)
Then use the rockets to move your whole columns over.
Make a match beside the cage under the pigeon, this destroys the mesh on it so you have only a metal and wooden cage.
Try to get the pigeon into one of the first two columns, both of which have line blasters (springs)
A line blaster can now be used to help clear the column and free your pigeon.
Wait until you have another rocket, then set it off in the same column that a blaster is in.
Because you're hitting it twice (with line blaster and rocket) it will clear both the metal and the wood from the pigeon cage in one move.