Simply put, the oxidizing agent takes electrons while the reducing agent gives them.
For instance, if you put copper metal in a solution of silver nitrate, you get this redox reaction:
Cu(s) + 2 AgNO3(aq) ---> Cu(NO3)2(aq) + 2 Ag(s)
Once you have a balanced equation of the overall reaction, the first thing to do is to find atoms or ions that have changed oxidation states.
The nitrate ions in this reaction didn't take or give away any electrons so they're not acting as an oxidizing or a reducing agent. You can tell because they stayed as NO3- throughout the whole reaction.
However, the copper metal lost 2 of its electrons to form Cu2+ (as copper(II) nitrate). Because it gave the electrons away, copper metal is acting as the reducing agent in this reaction.
Cu(s) ---> Cu2+(aq) + 2e-
Meanwhile, the silver ions in solution took these electrons and neutralized their 1+ charge to zero, forming silver metal. Since the silver ions are the ones taking in electrons, they're acting as the oxidizing agent.
Ag+(aq) + e- ---> Ag (s)