The video shows several ways of completing this level.
Method 1: No trapped Pets
Try to keep the cages in a line as much as possible.
Blasting the first lot is fairly easy (just drop the line blaster 3 blocks down), but once that is done the game gets a whole lot more complicated!
As soon as you remove the cages completely, more will drop.
So if you remove the metal with a line blaster then drop the plain wooden cages below the pets (keeping at least one block between them) it will be longer before more boxes drop.
If you remove them completely with two line blasters more metal/wooden cages will fall immediately, and it makes things a lot more difficult.
Position the line blasters using matches, other line blasters and the built-up hammer booster from in the game.
Matching blocks to move the blasters to where you want them can be tricky.
The example here does not use any extra boosters, but if you have block busters (hammers) they can be very helpful!
Block busters could be used to both position the blasters and to line up the cages.
Try to get the first 4 pets down all together, then deal with removing the metal off the 4 boxes that have dropped, and get another 3 or 4 pets to the bottom.
Then, for the last drop, you'll only need to save 2 or 3 to complete the level.
Method 2: Cage the Pets!
Another way of playing this Level is to use the cages to collect more Pets on the way down.
Let the Pets get trapped and another 4 Pets will fall, then release the trapped Pets with line blasters and drop all 8 of them down.
Now you'll only need 2 more, you can even do the same and get them using just one column
Or, of course, you can use a combination of both methods.