A continuity correction is made when we use a continuous distribution as an approximation to a discrete distribution, or the other way around.
For example, when we approximate the binomial (alive-dead) model with the normal distribution.
Page 18 of chapter 10 gives a good explanation of this, and also the example starting at the bottom of page 31 of the same chapter. On page 34 (same chapter) there is a good example of when the continuity correction will not be made (i.e. when it will have no real effect.)
When it comes to grouping ages with less than 5 expected deaths, one should especially do this when there are more than one age group with expected deaths less than 5.