Disclaimer: I've learned to use mirroring in meetings and negotiations. Sometimes I use it, sometimes I don't. It's definitely been used on me. I think it has some effect but I'm not sure how much. I'm a skeptic so I'll put my devil's advocate thoughts in as well.
Why might it work?
Mirror Neurons on Wikipedia has over 100 citations, many are scientific articles.
A mirror neuron is a neuron that fires both when an animal acts and when the animal observes the same action performed by another.13 Thus, the neuron "mirrors" the behavior of the other, as though the observer were itself acting.
But mirror neurons have not yet been shown to be responsible for the psychological effects ascribed to mirroring.
Why do people think it works?
Postural Mirroring and Intergroup Relations (Marianne LaFrance, 1985)
Both results are interpreted as evidence that postural mirroring is an obvious yet unobtrusive indicator of openness to interpersonal involvement.
But that doesn't necessarily mean that it makes other people more open.
What reactions might people have?
When It's an Error to Mirror: The Surprising Reputational Costs of Mimicry (Kavanagh et al, Psychological Science, 2011)
Mimicry and imitation can facilitate cultural learning, maintenance of culture, and group cohesion ...
Mimicry in dyadic interaction can be observed by third-parties, and the conclusions these observers draw have consequences for one's reputation in their eyes. ...
In particular, if one mimics the wrong individual, observers could judge one less competent than if one had not mimicked. As a result, mimicry may have reputational costs that at times make not mimicking one's partner a superior social strategy.
Does mirroring "work"?
Pentland, Alex. "Social dynamics: Signals and behavior." International Conference on Developmental Learning. Vol. 5. 2004.
In this paper I develop an automatic measurement method for quantifying some of these non-linguistic social signals, and describe how these measurements can be used to form powerful predictors of behavioral outcome in some very important types of social interaction: getting a date, getting a job, and getting a raise....
When one participant displayed mirroring behavior, the other would usually join in (74% of the time). ...
For low-status participants the use of mirroring was most important.
So mirroring might get you want you want but might also lead people to see you as low-status.