There are two steps required.
First, you need to download a Json file that contains information on the assignment. You can do so by going to the assignment, and selecting Download Assignment Data.
Second, you need to get the Python CrowdGrader assignment downloader from GitHub.
You can then run the downloader via:
./download_assignment.py <assignment_file.json> <destination_directory>
This will produce a directory in <destination_directory>, and it will create one sub-directory for each submission. In the sub-directory for each submission, you will find the submission metadata, the submission text, the files that were submitted as attachment, and the reviews. The format should be self-explanatory.
The reason you have to go through this two-step process is that an assignment can consist of a very large amount of data (up to several GB for some assignments, depending on the files that were submitted as attachments). Therefore, downloading an entire assignment can take a long time, and cannot be performed as a single download.
The downloader code is supplied with a MIT license, so please feel free to improve it and to share back your improvements.