If you want to find out about the the relationship of this response to romance in particular, there's a pretty comprehensive research paper called Love is more than just a kiss: a neurobiological perspective on love and affection which reviews a lot of the work done in this area. This is pretty good as it discusses all the different stages including breakup.
Essentially, there seems to be a lot factors at play which elicits various responses through different stages of love.
In terms of anxiety and stress, initially, there's a large increase in levels of cortisol and a marked drop in serotonin. Depletion of serotonin is found in many psychiatric conditions such as depression, anxiety and panic disorder. There's even been evidence that these levels at early stages of love are similar to those suffering with OCD. Obviously, the 'symptoms' of early love are quite similar to these conditions so it's hypothesised that this, at least in part contributes to those feelings.. It was found that these levels are back to normal in 12-24 months.
The elevated levels of cortisol contribute to the feelings of stress but this hormone has also been shown to promote attachment..
There are also changes in activity of the amygdala. Responsible for regulating a lot of emotions such as fear and sexual drive, the amygdala can activate autonomic nervous system responses through the mechanisms which Kevin described.
There are a bunch of other things at play here including dopamine, oxytocin, vasopressin, nerve growth factor (NGF), testosterone, increased activity in the hypothalamic-pituitary-axis (HPA) and variations in activity in other areas of the brain.
Again, that first reference gives a pretty comprehensive overview of the mechanisms involved.
To answer the actual question about the fluttering feeling, as Kevin mentioned, it's hypothesised that this is caused by reduced peristalsis in the intestines as a result of sympathetic stimulation. I don't have an academic reference for this but there are plenty of non-academic sources that talk about it (a quick Google of butterflies in stomach and physiology will probably show them).
A lot of the mechanisms noted above can contribute to, sensitise for or cause sympathetic activation. For example, panic attacks and anxiety, mentioned earlier with the serotonin relationship, can cause the same feeling.
Boer, et al. Love is more than just a kiss: a neurobiological perspective on love and affection. Neuroscience. 2012. 201:114-124.
Marazziti, et al. Hormonal changes when falling in love.
Psychoneuroendocrinology. 2004. 29(7):931-936.
Guyton and Hall. Medical Physiology. 11th ed. Elsevier Saunders.