How should I mention the period between the end of my post-PhD and the beginning of my work as research staff? I finished my post-PhD, after two years working under a scholarship, in April 2015. In September 2015 I was notified that I was accepted as research staff in my institution (I applied in December 2014). In September 2016 I was finally added to the "payroll" as a staff member.

Between the period April 2015 - September 2016, I kept receiving my post-PhD scholarship in what is known in my country as an "extension of the post-PhD". This is done because the time that passes between the end of the post-PhD and the final addition of your name to the payroll (if you are accepted) is rather long. In my case it took about a year and a half (which is the standard period), and this way one is not left with no monthly income.

The question is: how should I describe that period in my CV? Anyone from my country will be able to tell immediately what happened, but I'm not sure how to explain this to people from other countries.

In rigour my post-PhD ended in April 2015, so I can't really say I was working on it for an extra 1.5 year. What I did for that period of time was what I do now: research. But I can't say that I started working as a researcher after April 2015, because my first actual paycheck as such didn't come until September 2016.
In the comments you said it is called an "Extensión de posdoctorado".  It is fine to list that phrase as your job title during the period.  It is reasonably clear to english speakers.

People reading international CVs expect to see unfamiliar terms and career progression.