Ubiquitous Language is a term used in BDD and DDD and just means 'the shared language that you build with non-technical people'. It largely involves using business terms as much as possible, but often you have to try and agree some naming when talking to non-technical folk to resolve ambiguities. UL is not a technical concept, it's a way to focus on how you're expressing concepts in conversation.
In BDD we use this shared language to have conversations with customers about requirements, in the form of examples. In DDD we use a shared language to have conversations with customers about business process and rules. This article is bringing those two approaches together.
The stuff at the end is showing how the internal domain modelling (class naming, method naming, argument choice) can be taken directly from the conversation you had with the customer when you've had that conversation effectively. The significance is that the 'modelling' effort happens less in the developer's realm and more in the collaborative realm of customer communication.
Disclaimer: I work with the author of the article, use these techniques, and profit from their adoption