How I described developer marketing in new hire onboarding at a previous company is that a developer platform gives someone the ability to customize their needs in a particular software product. And there’s both a business and consumer opportunity to this.
For example, at a previous company we were buying email delivery software and I met with two market leaders. Both did about 80% of what I wanted, because they were built for common use cases, but my organization had our own way of doing things. So neither product completely solved our needs. A developer platform on top of either would have given my company the ability to make some customizations to solve 100% of our needs.
Another example is when you as a consumer use Facebook to log in to a website. With this functionality, the website owner doesn't need to create a login experience and also doesn't have to tap into the entirety of Facebook.
The Developer is the person that writes this code for a software product. Developer Marketing is how we as marketing professionals excite developers to try new functionality as well as evangelize the work that developers have done to excite customers to try these new user experiences.