Okay. You are off to a good start. Rupert Miller Jr. in his book Beyond ANOVA:Basics of Applied Statistics published by Chapman and Hall 1997 as a reprint of an earlier publication, he points out in fairly simple language the assumptions in ANOVA and what happens when there are departures from the underlying assumption and this includes specific robustness properties. So it may be that if ANOVA is robust enough in your case, you don't need to worry about the non-normality and so do not need to transform the data.
But in your case you know that people fall asleep sometimes while they should be reacting to a stimulus. It would seem that the observer should be able to note when this happens and you can eliminate the cases when this occurs. If this is the case you do not need a statistical technique to identify and remove the outliers. You then need to be satisfied that what remains provides a an adequate sample size for the ANOVA. Also you want to be convinced that the people you eliminate for sleeping would not have very different reaction times compared to the ones that remain.
Some comments about outliers: I have a lot of experience doing research on outliers and have also applied outlier detection methods when validating Department of Energy data bases. I think it is important to distinguish between outlier detection and and outlier rejection. Outlier detection helps you identify when observations departs significantly from the bulk of the data.
I view outlier detection as being acceptable since it leads you to check for a possible errors or other causes for the outlier. If the error can be confirmed, there is justification for removing it.
Detecting outliers in some situations can tell you a lot about possible problems with your experiment. But arbitrary removal is dangerous because you are forcing the data to conform to some preconceived notion as to what your data should look like.
As you become more comfortable with these subjects do look at some of the posts here about ANOVA, transformations and outliers. Also wikipedia can often be helpful as well as some elementary books and articles on outliers. Besides ANOVA , Miller's book deals with transformations, and one sample and two sample problems.