09 Yamaha 250 V-Star bogs down when throttle is twisted

Backfiring is usually a sign of the engine running rich - unburnt fuel is passing into the exhaust system.

Assuming it's a new carb for that exact model bike, I would assume it needs to be synchronized.

This can be done with vacuum balancing kits like this. (Link to Amazon)

If this doesn't resolve your issue, check your air filter. If it's plugged up and not letting enough air through, this will also lead to a rich condition.