Everyone likes feeling important
The biggest reason to give up the spotlight on occasion is because you probably like being in the spotlight. And as backwards as it might sound, giving it up on a regular basis is advantageous in the long run.
If you're old enough to remember when "multiplayer" involved multiple people sitting in front of a television, you probably remember the debate between Player 1 and Player 2. Consoles had two controllers plugged into the front, and Player 1 was the seat of joy and wonder and glory while Player 2 was a human wreck fit only for scorn and abuse. Everyone wanted to be Player 1, no one wanted to be Player 2.
My youth consisted of two systems for determining who got to be Player 1. The first was a simple arrangement: Whoever was Player 2 for this game was Player 1 the next time. If there were more than two of us there, it was a rotation of people. The second system involved two children or more racing through a house at the speed of sound, followed by a general brawl over whomever grabbed the coveted first controller.
Guess which system was more pleasant over the long run? People who stepped away from the Player 1 spot had to play less, but in the long run those were the kids you wanted to play with. Sure, you could get more playtime in if you completely dominated the big chair, but eventually everyone else decided that it wasn't worth bothering and you didn't get to hang out with other kids any more.
The comparison is pretty obvious. If you have to be the center of attention at all times, people will stop roleplaying with you, because you're treating the whole thing as your own personal show. Step away from the spotlight and everyone is happier to let you have a day in the sun.
When do you give up the stage?
While we're all a bit lost in our own little worlds (which is kind of the point of roleplaying), most of us don't specifically set out to exist in an echo chamber. The question is when it's time to be the bit player to someone else. It can be remarkably hard to gauge when you're simply acting like a wallflower and when you're politely stepping aside.
First of all, ask yourself whether or not your character has any skin in the game in the first place. If you're tagging along to help another character investigate a ruin, you might have a bit of a character moment, but your character is not the star of the show. On the other hand, if you were brought in specifically as a consultant for those ruins, you've got a bit more of a role.
Second, remember that you don't need to be the center of the room to have interesting things take place. One of my characters recently attended a major event in which she was a sideline, barely even present in the eyes of most of the guests and certainly not the guest of honor. But she ran into two people whose presence had a huge impact on her, despite not even being the focal point of their evenings. That led to a surfeit of roleplaying with several other people as a direct result, giving her time in the spotlight specifically because she wasn't the focus then.
Third, keep in mind that the supporting characters often wind up making a bigger impact than you think. Some characters don't do well in the shadows, but some are significantly bolstered by being the person everyone's met but no one really focused upon. Spies, yes, but also merchants, tradesmen, socialites, and general introverts. Later on you can re-connect with people, establish solid relationships, but it's all right to not be the focus in a huge scene. And there are less demands on your attention anyway.
But how do you pass it over?
Simple. You ask a leading question and then you shut up.
A lot of people get the first part but not the second. For that matter, they get the idea of a leading question but don't bother thinking about what that leading question will be. Someone roleplaying a simple miner is not going to be drawn into a discussion about politics no matter how many leading questions you ask, for the same reason that a politician won't respond to your questions about deep-shaft mining.
It's all a matter of taking turns. If you're in the spotlight, step out, pull someone over, and then stay out for a while. It's not going to hurt you, I promise.
Feedback is welcome as always in the comments or by mail to firstname.lastname@example.org. Next time around, let's talk about keeping a character fresh. The week after that, it's time to discuss being part of in-game organizations while fitting well with lore.
Every Friday, Eliot Lefebvre fills a column up with excellent advice on investing money, writing award-winning novels, and being elected to public office. Then he removes all of that, and you're left with Storyboard, which focuses on roleplaying in MMOs. It won't help you get elected, but it will help you pretend you did. If you need a refresher, check out the Storyboard Library.