As a GM, how do I encourage PCs to share and play out their backstories?

Rather than building a big back-story during character creation, you could ask questions of the players as they come up in play and then utilize the answers.

When they make a skill roll, ask who taught them that skill. When they cast a spell, ask them if their master taught them that spell or if they stole it from their master's spellbook when they weren't looking. When they don't trust an NPC, ask if they remind them of someone from their past who betrayed them.

Take down notes as they answer and make sure their answers have meaning. NPC's could ask questions, from a merchant having tea with the PC wanting to know what their parents did as a trade or a child wanting to know what kind of games they played as children in the lands where they grew up but don't be afraid to ask as a DM to the player. If they don't have an answer right away, ask them to think about it and come back to it. Don't let the questions stop play or cause bottlenecks that lead to pressure.

The most important part, as noted above is that the answers have meaning and effect play and become part of the world.

I'm going to take the liberty of assuming that your players are creating their characters in isolation from one another, and therefore simply end up slapping mechanics together without any thought of why the character is what he/she is.

I would suggest that your first session of the game actually be a character creation session. Don't think of character creation as something you do before play. Think of it as just the first part of play. Have the players create their characters together, with you guiding them along. Ask questions. Lots and lots of questions. Feel free to make them leading questions, even.

• "Where did you learn to fight like that?"
• "Who taught your magic?"
• "What did you do before you were an adventurer?"
• "What was the last adventure that you two had together?"
• "That lute you carry. Who owned it before you?"
• "Your from the same place. You must have mutual friends."
• Etc.

Point is, you want your players to think about their characters as more than just their mechanics. Find interesting tidbits about the characters and poke at them until they yield interesting ideas. Ideally, doing this in a group setting will allow all your players to bounce ideas off of each other, coming up with more interesting details than they would on their own.

I suspect that players who have problems with that think of the characters they play too much as tools or avatars of themselves, and not as characters different from themselves.

The first step would be to point this out to the players and explain that in the interest of a more exciting game experience they should try to flesh out their characters.

In order to get the ideas gowing, it can help to give the players a list of questions to think about:

• Where and how did the character grow up? Why and how did they leave their home?
• Does the character have living relatives? If yes, how is their relationship? If no, how did they die?
• Has the character had romantic relationships, and how did they turn out?
• What's the character's goal in life, what does he dream of achieving?
• What is the character's greatest fear and where does it stem from?

Anime is a good source of inspiration for this. Because of the length of a show, they can (and do) delve into the back stories of many of the secondary characters. Generally, the story has an impact on the current story arc but not always. A character form their back story comes and seek revenge/help/interacts and flashbacks occur. Those can help flesh the story, giving it a feel of information is delivered when it is needed to understand the story. The alternative to this is foreshadowing where that information is already there but maybe masked or seem irrelevant. So, you can look at anime like Ghost In The Shell: Stand Alone Complex (both seasons), Cowboy bebop, and Fullmetal Alchemist -- to name but a few good ones and see how they do it.

In general, doing flashbacks within a RPG, keep in mind:

• Make sure your players know and understand what key plot points you wish to have in the flashback. For example, Bob dies killed by Kate.
• Make sure the players can introduce interesting story elements at this stage into the plot. For example, Kate is revealed to be $PlayerOne's sister. Note that this means that you may have to re-write large chunks of the plot.
• The players are not there to win. They are there to build a more interesting story. They "win" in the game, not the flashback. For example, Kate will escape alive, maybe saved by her brother?
• The flashback has to be short. You are distracting from the main story line. The point of the flashback is to add information to the main story, not tell its own story.
• As a referee, say yes. Take on-board things that the players come up with.
• As a player, stay true to the thematic content of the main story. Try to enhance it rather than try to gain power from the flashback.