Studying child development is important to teachers for a number of reasons.
First, it provides a context for the stages of development; teachers need this base of understanding in order to develop appropriate curriculum choices for their students. This is why, for example, it doesn't make sense to attempt to teach algebraic concepts to first graders; in most cases, they have not developed the ability to think in such abstract ways. All students will not progress at the same rate, and understanding how the stages of development progress will help teachers modify curriculum choices for students who are struggling or excelling.
Understanding child development also helps teachers as they assist parents in identifying potential concerns in the learning process. If a ten-year-old has not yet developed the ability to understand the perspective of others, he may not have progressed beyond the pre-operational stage. If a fifteen-year-old cannot convey ideas with sound logic, then they may not have progressed beyond the concrete operational stage. Students who fail to develop as expected may need additional educational supports in order to find academic success.
Studying child development can also help teachers assess the source of a student's social or behavioral problems. A student who avoids doing assignments may not just be lazy; he may be unable to complete the work as it has been assigned. It also helps when comparing a child against her peers; teachers who understand the fundamentals of child development will understand that children are all on different continuums of progress, and being "behind" at any given moment does not imply an eventual negative educational outcome. It does give teachers tools for evaluating a child's needs and then targeting ways to encourage further development so that a child reaches her best potential.
Finally, having a thorough understanding of child development helps teachers evaluate new directives, methodologies, and curriculum choices made by those outside their classrooms. Often, people with little to no educational training try to influence the decisions that teachers make with their students. Teachers who have studied child development are better equipped to assess these changes and to then determine which, if any, are relevant and appropriate for their own students.