How do birds influence brood sex ratios?
On a more general level it is possible for parents to influence sex-ratio of broods in a number of ways, and there are other reasons why a brood might have a sex ratio distortion. The paper I've been reading this morning is about side-blotched lizards doing just this. I'll give a quick basic answer just from the top of my head and hopefully someone can give some specific examples in birds.

Bias in gamete production

Assuming XY chromosome systems where the males are heterogametic (XY, females XX) the male can influence the brood sex ratio by producing more or less Y-bearing sperm. Sperm carrying an X-chromosome will always produce daughters and sperm carrying a Y- will always produce sons. The opposite would be true for ZZ/ZW where female gamete type determines the sex, they could only produce Z eggs, therefore only having sons.

Female cryptic sperm choice

A fairly modern idea is that females can selectively fertilise eggs with sperm based on the DNA they carry, this could include whether the sperm is Y- or X-bearing. This idea is thought to explain how females produce daughters from small sires and sons from large sires in the paper I mentioned at the start.

Meiotic drive and sex ratio distorters

Meiotic drive elements may be responsible for sex-ratio distortion in broods. Just like the biased sperm production this can alter the sex-ratio at the gamete level. Genetic effects mean that fewer offspring of one sex than the other are produced. Here is one example where suppressors of the distorter have evolved in the autosomes.

Direct selection by the parents

In the paper you mention above, and with most studies, it is possible that the parents are selecting which offspring to raise after they hatch (or as an egg - it may be possible to determine sex at that point) and if females are more costly to raise and conditions are poor, they may kill/neglect the female offspring early on.

Within brood competition 

If one sex is stronger as a chick it possible that the broods could become sex biased in certain conditions which consistently favour one over the other.

Consequence of environmental strain

Similarly to the last two points, if conditions are poor it may be harder for one sex to reach maturity than the other which would result in sex-bias. This is the conclusion the authors in your paper seem to come to.

Environmental effects

As with some reptiles, environment can determine the sex of the offspring. For example, incubation temperature in turtles has a considerable effect on offspring sex ratio. Given how close birds and reptiles are in evolutionary time, and the importance of active incubation in birds, I don't see that it's impossible for the parent to vary some kind of environmental factor in order to bias sex-ratio.

There are certainly more potential mechanisms and some of these ideas are fairly new in evolutionary biology research so there are many unanswered questions. I hope I will have time to find some bird specific examples later.