Download the persist-message.def, persist-message.lua, query-message.def and query-message.lua files and put them in a plugins folder on your filesystem
Create a persistence folder on your host system. In this folder the docker container then creates the sqlite3 database.
Adapt your docker-compose.yml to enable the plugin and map the required resources into the docker container

services:                                                                                                                                                                                                        
  signal-cli-rest-api:
    image: bbernhard/signal-cli-rest-api:latest
    environment:
      - MODE=json-rpc #supported modes: json-rpc, native, normal (choose the mode you want; the plugin works with all modes)
      - ENABLE_PLUGINS=true # enable plugins
      - "./plugins:/plugins" #map "plugins" folder from the host system into the docker container
      - "./persistence;/persistence" #map "persistence" folder from the host system into the docker container
      - RECEIVE_WEBHOOK_URL=http://127.0.0.1:8080/v1/plugins/persistence/persist-message #register an internal webhook endpoint

Restart your docker container

Every message that is received is then written to the messages.db inside the persistence folder.

The stored messages can then be received via the REST API with:

curl -X GET 'http://127.0.0.1:8080/v1/plugins/persistence/query-message'

Debugging and Troubleshooting

Make sure that the docker container has write permissions to the persistence folder
On the host system, check if the messages.db gets created in the persistence folder
Check the logs. Do you see any error?