Why don't action and reaction forces on a person swimming in water cancel each other out

They Act on Different Objects

Explanation:

Newton's Third law of motion states that action and reaction are equal and opposite. The action and reaction forces in of Newton's Third Law -works on different objects.

In Real life examples, or situations  like, A textbook lying on  a table  ever, A cup pressing down on the table, or a table pressing up on an iron, these forces in this cases,  are balanced by extra or additional  forces, and at times, they are not.

Most importantly  action and reaction forces from Newton's Third Law of motion act on different objects, But  forces working  on the same object or entity  may cancel  out this is from( Newton's Second Law), So long as the sum of their is considered  to be zero.