Baking Immutable Images
An idea key to your question is that of immutable infrastructure, which is the idea that an image is built once, deployed many times, and never changed at runtime. If the contents of the image need to change, an entirely new image is built from scratch and new instances replace the ones running the old image.

This is the opposite of what you might now call mutable infrastructure, where often a machine (or container) is booted from a generic OS image and then its contents managed by a configuration management tool on an ongoing basis. In this model, instances generally live longer and are changed in-place to respond to new requirements.

Packer is intended as a building block for immutable infrastructure. It is intended to orchestrate the creation of customized images that boot immediately into the desired state. If immutable infrastructure is the right fit for your use-case (which is not necessarily the case!) then indeed tools such as Ansible, Chef and Puppet may be unnecessary.

However, running these tools in a "one-shot" mode (e.g. Chef Solo) with Packer can often be useful for non-trivial images: these tools generally provide more powerful abstractions than Packer alone provides, making it easier to do things such as configure the init system to run certain services, or create user accounts. In this case, the configuration management tool would run as a provisioner in Packer, creating the necessary configuration within the image, but is not involved in the ongoing upkeep of instances booted from the image.

Most practical systems include a mixture of mutable and immutable infrastructure, since e.g. immutable infrastructure is inappropriate for stateful, centralized systems. In that case, using a traditional configuration management system can be useful to create a common base of functionality between your immutable and mutable systems, applying configuration in a one-shot way for immutable images and in an ongoing, stateful way to long-running, mutable machines.