This is one where I'm especially looking forward to what other people have to say - it's something I want to be good at, as a player, so I'm hoping to pick up tips.
For me, at the moment, I think it probably boils down to the way you interact with the world. You can accept the quest, slaughter your way through the undead-filled dungeon, retrieve the macguffin, and return to receive your reward. So far, so computer game. Or, you can explore how you know the quest-giver, what you know about them. You can befriend them (and not just because a quest-giver is a useful person to have on your list of contacts). Then in the dungeon, as you slaughter you can investigate why it's so filled with undead (which might lead to more adventure hooks). If appropriate, you can make the macguffin more personal to you, too. So, if you're rescuing someone, talk to them, find out about how they ended up kidnapped, find out where they're from. Return to the quest-giver and don't just receive the reward (or argue with the quest-giver in the hopes of receiving a bigger reward just for the sake of it - that's one character interaction I see often and find boring), but recount tales of your heroism, or otherwise continue the interaction you started before you left.
In our Mage game, Howard and Ragna stole an RV and driving from Los Angeles to Chicago (to get holy water to fight vampires). Howard is LA born and bred, while Ragna is from Norway originally and has only seen LA in the US. Ragna's the better driver, but doesn't have a valid US licence. That didn't seem as important in a stolen RV, so they were driving in shifts. It was late evening, early night, and Howard's shift was ending. He pulled over on the side of the road at a safe spot, to stop for the night. It was eerie - tall pines grey in the gloaming, darkening fast as heavy clouds came over. The headlights picked up silver threads of rain.
A sudden thud against the RV woke Ragna with a start - she looked round to see Howard even more startled. Even his cat, Marcel, seemed disconcerted. Another rattling thud shook the truck. The storm didn't bother her, but Howard's fear spread through Ragna: she'd never seen him show any signs of it before. His face was white with it now and she wondered what he'd seen. She peered through the windows, but couldn't make anything out in the blackness beyond the pelting rain.
The next rattling was accompanied by an unearthly howling, which snapped Howard into action: he leapt back into the driver's seat and Ragna buckled in beside him, trying to calm her companion as she brought up a google maps route to the nearest official truck stop. Once there, he stepped outside and smoked an entire packet of cigarettes.
It was just a storm, but Howard had no idea what a storm was like outside a city. Ragna knew very well, but was infected by his fear. Our GM afterwards said he'd half-intended to have us attacked by werewolves, but the way we engaged with and expressed the experience meant he changed his mind.
So that's my advice: get curious about the world. React to it. Use the scenery. Make friends with the NPC's. Believe in it.
If you're taking part this year, I'd love it if you'd include a link to your reponse in the comments below. Look forward to hearing from you!