The paper controlled for maternal effects - and it comes from the massive Millennium Cohort study so it is large (9000 mother-baby pairs) and perfectly respectable.
What is very hard to assess is 'reverse causation' - like Gnome says, what if the 'quieter, calmer' babies are the ones who are easier to breastfeed? So it's not breastfeeding that's 'caused' the lack of behaviour difficulties, but the lack of behaviour difficulties that has 'caused' the breastfeeding!
So they come out of the uterus already 'easier' to breastfeed - and we know that there is a difference in stress levels in babies linked to stress in pregnancy, so this is not a mad idea.
You'd have to show that 'stressed' babies are less likely to be breastfed in order to take this idea further, and I don't know of any work that shows this. In fact the opposite could be the case - that breastfeeding becomes an easier way to calm a stressed baby.
No one needs to feel bad as an individual about this, though, because none of it can be taken as an indication of what your baby/child would have been like. The research of this type is less an 'instruction' to mothers, and more of an encouragement, and it is also a spur to public health measures to make it easier for mothers to breastfeed for longer.
The stats can be translated like this: take 2 school classes of 30 5 year olds. One (A) has kids in it who were all bf for four months or more. The other (B) has kids who were not bf for four months or more. In A, there will be 1 or 2 kids with behaviour problems (as defined in this paper). In B, there will be 4 or 5. In both classes, the maj. of the kids are fine.
So indivdual terms the diff. is not huge, but in public health terms, this is quite significant and certainly supports existing and other strategies to enable more breastfeeding.