Run docker-compose up to launch the containers.
Note that you'll see lots of output as the various software launches in parallel.
$ docker-compose up
One the output from compose stops, in another terminal confirm you have the containers up and running.
$ docker ps ~
CONTAINER ID IMAGE COMMAND CREATED STATUS PORTS NAMES
7af731797a95 puppet/puppetexplorer "/usr/bin/caddy" 2 minutes ago Up 2 minutes 0.0.0.0:32828->80/tcp compose_puppetexplorer_1
8f8aecd2cc69 puppet/puppetserver "/opt/puppetlabs/bin/" 2 minutes ago Up 2 minutes 0.0.0.0:32827->8140/tcp puppet
1fefb0ea22ff puppet/puppetdb-postgres "/docker-entrypoint.s" 2 minutes ago Up 2 minutes 5432/tcp postgres
910002aa3e6a puppet/puppetboard "/usr/bin/gunicorn -b" 2 minutes ago Up 2 minutes 0.0.0.0:32824->8000/tcp compose_puppetboard_1
78bfd7967680 puppet/puppetdb "dumb-init /docker-en" 2 minutes ago Up 2 minutes 0.0.0.0:32826->8080/tcp, 0.0.0.0:32825->8081/tcp compose_puppetdb_1
Note the port number for the puppetexplorer container, in my case 32828.
If you're running docker locally you should be able to access the dashboard on 0.0.0.0:32824 (your port will likely vary).
If you're running docker-machine then you can run docker-machine ip to determine the correct IP address.
Using this you can access you puppet explorer dashboard.