Like most aspects of performance, the best way to get better at climbing out of the saddle is to do more of it. A simple session for doing this is to find a hill that takes 10-15mins to climb and, having warmed up riding to the hill, do reps where you alternate 1min seated with 1min standing, ensuring that you shift up at least one gear when you stand up. If using a power meter, you can do this workout as an over/under.
For the out-the-saddle minutes, aim for 70-80rpm and 110% of FTP; for seated, spin up to 80-90rpm and 90% of FTP. Coast back down to recover and repeat 2-3 times.
Unfortunately, working on out-the-saddle climbing on the turbo won’t be anywhere near as effective. Part of being able to climb strongly is trunk stability and strength, which simply isn’t challenged enough on a static indoor trainer.
However, you can definitely make a difference in the gym. Press-ups, renegade rows, planks and oblique planks can all help boost strength.