To do something gamely is to do it in an enthusiastic, plucky, and determined manner. If you study vocabulary gamely (and we know you always do), you make it a point to have a great time learning new words.
When you're game for something, you're excited and ready to do it. Likewise, to act gamely is to have a positive, upbeat approach to something. If you gamely agree to help your friend cram for a test, you might show up with snacks and good music — just to make things a little more fun. When you behave gamely, you're bringing good vibes. A coach who runs practice in a gamely manner is likely to inspire confidence in the team.