Hip, I dropped ~10 pounds between the start of March and the end of April this year and in part I think it's made me a much better climber than I was a year ago.
It wasn't just dropping the weight, though. I lost the weight by working hard on the treadmill, trainer, and then outside. Riding up steep hills more often helped, as did getting my tail handed to me on a weekly basis by faster riders.
So, yes, not having to haul up more kilos helps in the climbing, but actually getting out and riding up hills (and then riding back up them a few times) helped just as much.
Also, don't underestimate actually knowing a climb. The more times you do a specific climb, the 'easier' it becomes (notice easier is in quotes there) through familiarization. Mentally knowing you can do it is half the battle. Before this year, I never got all the way up Blue Mound without walking part of it. This year I pushed myself the first time I was there and once I made it up, I knew it could be done. Subsequent visits up there have been much 'easier' mentally, even though the road is still as steep and long, you know?
Get out there an climb. And do it again. Etc.
YOU CAN DO IT!