To move the Pet across the board you'll need to slide the columns.
For a column to slide the column to it's left must be removed completely.
They can be removed with matches (if all blocks in the column are the same color)
This can be done easily at the beginning of the Level because of the locked blocks stopping more from dropping.
Another way to remove a whole column is with a rocket, the in game booster for this level.
Before you shoot the rocket make sure there are no mesh covered blocks (that will not go in one rocket swipe) as that would mean the column is not completely gone!