Whether they realize it or not most players will form an attachment to some aspect of their character, something their character has, in a game. As the game progresses this attachment will grow. Sometimes it can even become a focal point for the character. Perhaps it is something mundane like a favorite hat. Perhaps it is something more serious like a favored NPC or a weapon that gives some nice bonuses. Whatever it is, you can often get the character to act in all sorts of interesting ways by threatening that thing.
Identifying The Thing
The first part to doing this is that you have to identify the thing in question that can be used. Sometimes this can be very easy (i.e. one of the characters in my L5R game is known to state "I love my wife" at numerous junctures) or it can be very hard (another or my characters doesn't seem to have any solid attachments at this time.) A lot of times this takes some guess work on your part as the GM, but that's ok too. Why? Because your guess work is also giving the character things to do. Before you do the big hit, do some smaller ones. Poke and prod at things in the character's possession and see how they react. Whatever gets you the strongest reaction is likely the thing you are looking for.
Providing The Thing
Sometimes all attempts to identify what it is will fail. When that happens, it may fall on you to provide the thing. Often times this works best as an NPC. Someone you can introduce to the PCs world and have be helpful. A lot of times though you want to find a way to do something else, because players can naturally be suspicious of new NPCs - especially useful ones - and thus keep some distance between their char and the NPC. Whatever you do, the trick here is to get the player involved with some aspect of your game in such a way that there is a positive response. Perhaps you hook them into caring about a restaurant by having them get free food there after saving the owner's son. That link can then be used to draw the character into the stories going on around the restaurant, until the player begins to identify it as "their" restaurant. At which point, you have them.
Subtle Threats Work Best
When it comes time to threaten things there are usually two options. You can be subtle and do a passive threat, or you can do a direct threat. While a lot of times the direct threat will get you faster results, I want to make a case for the subtle threat. Let's take a look at the restaurant example. You've got your player to care about this restaurant through whatever means. That is our baseline assumption.
The Direct Threat
The owner's son is kidnapped and the owner comes to the PC for help. As this is a threat to something the player identifies as "theirs" in the game sense, the PC will help and goes out to recover the son. This can lead in to an investigation, probably an action sequence or three, and then the PC returns with the son and all is well.
Sure this idea works well, but it is also very much an over and done sequence where the player is reactionary in almost all steps along the way. However, you could go a different route:
The Subtle Threat
The player comes into the restaurant to find a couple of broad-shouldered goons in nice suits talking to the owner. If the PC looks around they'll notice that several of the windows in the restaurant are new, as well as the glass piece behind the bar. Money changes hands and the goons leave. If the player asks, the restaurant owner is being extorted for protection money from a mafia. The player now can choose to get involved or not, but all the onus for that is on them.
See the difference? In one, the direct threat, the player can almost feel forced into the situation. After all, if they don't do anything then a kid is going to be harmed and that is probably going to put a stop to the free meals they've been getting. In the other, the subtle threat, the player has the choice of getting involved or not. The instigating event is on them if they want to do it or not. Even better, since the player chooses to go in or not, they will very likely be the one acting instead of reacting. Granted, that may not be the safest course for your city/restaurant, but the PC will be the one setting the tune.
If The Player Doesn't Go For It
The problem with being subtle to light fuses under your PC's asses is that sometimes they won't go for it. Maybe the Player doesn't want to get involved with the shakedown story line so they ignore it. That's fine, and it is on them. Ultimately the player is making a choice and telling you they don't want to be involved in that. However, that also means that if they feel "nothing happened" for them that session that it is on them. You provide opportunity, no matter what the case, it is their job to play into it or not.
Don't brow beat your players into following up on plots if you can avoid it. Just poke, prod, and when you find something that the player will act to defend, dangle it over a fire and watch what they do. It's probably going to be a lot more entertaining than anything you could plan on your own.