Every easy board falls to four moves, and each one needs you to look at only two or three squares at a time.
Two alike side by side. If two cats sit next to each other in a row, the squares at each end of them must be dogs. A third cat there would make three in a row.
A gap between two alike. Cat, empty square, cat: the empty square must be a dog, for the same reason.
The signs. An = between two squares means they hold the same animal, and × means they hold different ones. As soon as one side is filled, the other follows.
A full count. Every row and column holds three cats and three dogs. Once a line has its three cats, every empty square left in it is a dog.
Work round the board with these four and each new animal opens up the next move. You never need to hold more than one line in your head. If you get stuck, check the signs first: on a busy board they are the moves people miss most.