(From a quick skim of the article, I'm not a fan of the analyses.)
Fisher's exact test is often ill used, in my opinion (cf., here: Given the power of computers these days, is there ever a reason to do a chi-squared test rather than Fisher's exact test?). Fisher's test is based on the assumption that that the marginal counts were fixed in advance and the only thing that was free to vary is which cells the observations fell into, subject to the constraint that they sum to the margins. It's hard to think of many cases where that even could be true.
The one paradigmatic case where it was true was the tea tasting experiment where Fisher used it. Fisher was at a garden party where a woman was complaining that if milk was poured into a cup first, and the tea was added afterwards, it ruined the tea, but that if it was prepared correctly (i.e., the tea was poured first, and the milk second), it was fine. Others scoffed at the idea that she could tell the difference. Fisher, rising to the occasion, offered a solution: they would prepare eight cups, four each way, and the lady would state which four she thought were which. Those data could be tested with Fisher's exact test, because there four of each preparation (specified in advance of the experiment), and the lady was obliged to choose only exactly four for each category of response. It turned out that she was right every time. That is (and this is the classic dataset for Fisher's exact test):
Preparation:
She said: M->T T->M Total
M->T 4 0 4
T->M 0 4 4
Total 4 4 8
fisher.test(table)
# Fisher's Exact Test for Count Data
#
# data: table
# p-value = 0.02857
# alternative hypothesis: true odds ratio is not equal to 1
# 95 percent confidence interval:
# 1.339059 Inf
# sample estimates:
# odds ratio
# Inf
There is a nice write-up of the event and Fisher's test on Wikipedia. (Let me acknowledge that I am going rather hard on Fisher's exact test here. In most cases the result—with respect to the significance—will be the same either way, and you could argue that it doesn't matter that the assumptions aren't met, but it typically has lower power than the proper alternative.)
Regarding the idea of covariates, I'm not really sure what is being referred to. Certainly Imbens and Rubin are famous statisticians, but I don't have access to the book, so I'm limited in deciphering this. At any rate, Fisher's test does not allow for covariates. If you had a binary response variable and some covariates, you would typically use logistic regression. Fisher's test is often used by people because they have very few events. If that were the case, there probably wouldn't be enough data to support the inclusion of covariatess, so the issue seems moot.
For what it's worth, you can infer the dataset from the article. In table 1, it lists the three conditions with their $n$s as: Honey (n=35); DM (n=33); Nothing (n=37). In the text above, it states that, "Fisher exact tests were used to compare adverse event rates between treatments". At the end of the results section it states:
Few adverse events occurred in this investigation. The combination of mild reactions that include hyperactivity, nervousness, and insomnia occurred in 5 patients treated with honey, 2 patients in the DM group, and no patients in the no-treatment arm (P = .04). In the honey group, the parent of 1 patient reported drowsiness and the parents of 2 patients reported stomachache, nausea, or vomiting, but these adverse events were not significant when examined separately from a statistical perspective (drowsiness, P = .65; stomachache, nausea, vomiting, P = .21).
Thus, I gather the data were:
## Patients:
Treatment:
AE: Honey DM Nothing
Yes 5 2 0
No 30 31 37
## Parents (drowsiness):
Treatment:
AE: Honey DM Nothing
Yes 1 0 0
No 34 33 37
## Parents (stomachache, nausea, vomiting):
Treatment:
AE: Honey DM Nothing
Yes 2 0 0
No 33 33 37
