Contact Form Extensions - Checkbox Templating
You're pretty close. In your input you have name="message[type][]". The little [] at the end there means means the data will be stored in an array, something like:

submission: {
  message: {
    type: ['New', 'Used']
  }
}


You have two options here:

Option one

You can change your inputs to be name="message[type]" instead of [type][]. This should result in a data structure more like you're expecting 

submission: {
  message: {
    type: 'New'
  }
}


Then your conditional check can be kind of like you have it, but this won't allow multiple values (not sure if you want those or not).

Option Two

Update your conditional check and output to properly output the array values. This would mean something more like:

{% for type in submission.message.type %}
  &lt;p style="font-size: 16px; line-height: 1.625; color: #51545E; margin: .4em 0 1.1875em;"&gt;
    Type of Quote: {{ submission.message.type }}
  &lt;/p&gt;
{% endfor %}