How To Introduce Limited Player Narration
I was introduced to this kind of play with Robin Laws' excellent Feng Shui RPG. It explicitly incorporates this kind of "light" player narration (as opposed to full co-authorship as is common in indie RPGs today). I still use it as training wheels to help people get into the mode of improvising. In Feng Shui, not only is use of tactically-correct battlemats contraindicated, but you're encouraged that if you are, for example, in a fight in a pizza parlor, to just say "I snatch up a pizza cutter from the table and cut the fool attacking me with it" without having to ask the GM or roll for the pizza cutter. If it makes sense and is genre appropriate it's fine.
Many people need that help to get started with the idea because most old school trad games definitely come from the paradigm that the player isn't allowed to contribute to the narrative at all - if the GM does not explicitly instantiate something, it doesn't exist. Which has some good attributes - too much player narration actually leaves me cold as a player since I prefer immersive character play - but some light player narration does make the game run more smoothly and be more exciting.
Just Say No!
In fact, in general I tell people they need to man up and improvise those kinds of things themselves because if they ask the GM I'll say no! "If you ask me how high the helicopter skid is, I'll say 'too high.' If you say 'I jump and grab the helicopter's skid,' I'll say 'roll it!' If you ask me if there's a pizza cutter within reach I'll say no..." That breaks them of the mother-may-I syndrome, which slows down the game. Having to say no once in a while is less time spent and more empowering.
Run a One-Shot To Introduce the Concept
With my current play group, I ran a one-shot of Feng Shui before launching into a longer campaign to train them on the mode of play. This gave them an opportunity to experiment with limited narration in a safe short mode environment. Then when we started a longer game in a traditional game system, the techniques port right over. You can use another game, just make sure it's yielding the type of narration you want - if you go too far into indie land then you'll get your normal D&D players saying "I declare the lich king is my long lost aunt and he's really after the strumpets on the moon!" (Fun fact: on Golarion the moon is full of strumpets, for real.) Which is fine if that's the kind of game you want, but that's not this kind of limited-narration environment which remains sim/immersion friendly while still getting a lot of the benefits of a narrative game.
Controlling the Narration
As for controlling the narration, we have very seldom had problems with that. We had the understanding that narration had to be very limited and appropriate - "belaying pin on the deck of a ship" is a valid pickup, "magical metal detector" is not - and the GM can always say no. Generally this is self policing, as "I come up from behind the pizza oven with a rocket launcher!" triggers everyone's cheese alarms and they just frown at the overexcited guy till he comes up with something less stupid (unless you're playing Toon or something very postmodern). All the "yes, but" stuff is fine for certain kinds of games, but it is unnecessary and disruptive for a game trying to be sim with limited narration only. A couple "no"s to set limits is much less hassle for everyone than trying to put conditions on inappropriate things. With adults it really won't come up much at all after some initial boundary exploration.
Mechanics and Limited Narration
Mechanics to dictate this are not needed and in fact I find them to be intrusive and harmful. The goal of limited narration is to be able to improv; if you have rules it's not really improv-ing. Whenever I have to shepherd currency (like FATE points) to do something cool, or make a roll separate from the roll I was going to make anyway to do a stunt (like the rolls for improvised aspects mentioned in the comments), that's lame because it 1) brings in more metagaming, taking away from character immersion and 2) takes away from the velocity of the game, which limited narration is intended to increase.
The one mechanical note I'll make is that people will try innovation to the extent that it's not punished - if you put big penalties on improvising it will tend to work against it. Feng Shui, as this is what it's about, went to the extent that you actually got a bonus to your rolls when doing a cool stunt. In d20 it's easy to rack up "-8 to do that because you don't have the 'slash a fool with a pizza cutter' feat" - make sure and encourage actions you want to encourage with the numbers. Removing a penalty or adding a bonus based on how cool something was is completely within your purview as a GM, despite the bizarre newthink from D&D 3e/4e that says it's not.
In D&D 2e, I had a halfway mechanic that worked - a Luck stat one would roll to see "if there's a pizza cutter". This worked in that campaign only because it was a very, very gritty low power sim game. I wouldn't use it in a normal power game and would just go to the "yep, if it's not dumb you've got it" model.
Though be careful of trying to over-mechanize it like Exalted does as described in @SimonGills' answer - I remember playing Roanoke, where you get a +1 per descriptive word you use on every action, up to +5. It led to some pretty sad results,as people just strung together tortured 5-adjective strings to get the bonus since the rules rewarded "5 words" instead of the desired result of "a cooler game." I prefer just "that's cool you get the +2 bonus" or not, when you start trying to over-define it it turns into a rule to lawyer instead of an improvisation to add to the story. Also, our stunt level is usually at their +4/+6 level - "I scream and hack at the orc" is expected in decent RP, not bonus worthy.