It implies a sense of:
* Taking a chance: You're not sure if it will work, but you're willing to try.
* Effortless action: It's not a big deal, just a quick attempt.
* Experimentation: You're open to seeing what happens.
Here are some examples of how you might use it:
* "I'm not sure if this new recipe will work, but I'm going to give it a whirl."
* "That job sounds interesting. I think I'll give it a whirl."
* "I've never tried yoga before, but I'm going to give it a whirl."