As crape myrtles pass their prime bloom season and begin to look a little “shaggy”, many people want to tidy them up to look neater through fall and into winter.
Removing the spent flower clusters can make the tree look nicer (though on a large tree this may be impractical), but doing so is entirely for aesthetic purposes. You may be tempted to continue your cleanup and get some pruning done while you're at it. Be careful you don't go too far!
At this point in late summer/fall it’s also OK to remove broken branches, or branches that have sagged and not sprung back due to the weight of flowers and rain. (The “pruning” practice sometimes referred to as crape murder can cause long, thin branches that are more likely to sag. Don’t do it!) These should be removed all the way back to the branch they originate from.
What you do not want to do is cut branches back partway, or remove too much overall at this time. This can encourage a spurt of new growth too late in the season to become mature enough to survive freezing temperatures.
The proper time to prune crape myrtle is late winter (January-February). Since crape myrtles form their flowers on new growth, there is no danger of removing already-formed buds.
For instructions on how to correctly prune a crape myrtle—and how to repair “crape murder”— follow this link: Pruning 101 - Easy crape myrtle pruning.