The attitudes, beliefs and philosophy surrounding the importance of design with regards to product, brand and marketing. Design culture isn’t relegated only to the design teams; it should be cultivated across all teams that have a role in helping your product succeed in market.
Here's an analogy.
Organizations always have a predominant culture, but within the company there will be several overlapping cultures that guide smaller teams. Much like you can have a culture for a region, a state, a city, or even suburbs within a city, a design culture is a part of a larger culture but one that the product team must adopt.
Why does it matter?
The digital product space is highly competitive. A decade ago, simply having a digital solution was enough to create a profitable business. But today those same businesses are competing against many others while consumer standards for user experience and design are higher than ever. Tech companies a decade ago were often engineering-driven, but today they must be design-driven. Design has to be important to everyone on the product team.
How does it apply?
Product managers, designers, developers, QA, customer support, and even marketing and sales have to move fast. They must iterate quickly, respond to customer feedback in-flight, and add in new features collaboratively. By sharing a strong design culture, they can ensure the digital product will maintain quality without slowing the teams down.