So Johnson in his speech (which is available in the video in the BBC article) is suggesting it is normal for a new government to take a break to set out a new programme of government (which is delivered in a Queen's speech, also mentioned in his speech).
Some people who see his viewpoint are happy to overlook the timing and don't necessarily see it as undemocratic.
There is another element. Direct democracy believers feel that parliament is trying to crush democracy (by potentially overturning a referendum result).
Representative democracy believers cannot accept that their representatives would be denied the ability to represent their constituents at such a critical time of major constitutional change.
So the dual narrative was a reference to how some people see this step as delivering the power and will of the people.
Others see it as a dangerous subversion of democracy.
Happy to clarify more if I'm being unclear but you can read the comment section on the BBC article and you'll see incredibly polar comments. The dual narrative is precisely that. People read the same article, see the same speech and have a completely different understanding of it.
Perhaps that's unremarkable to you, but in the UK it's been profound and profoundly different from the mainstream politics I've seen my entire life.
[Edit - clarification of crush democracy]