This forum has set the rule for questions to be twenty years somewhat arbitrarily (as discussed here), and this doesn't necessarily reflect what professional or amateur historians study. But it would be quite hard to engage deeply and historically with unfolding events: I'm sure you can think of events in the last decade that were really important in their immediate aftermath that, after some distance, aren't (or reflect wider trends/are particular aberrations and don't reflect those trends); the converse also holds true.

The collapse of the Soviet Union, for example, is outside the bounds of the twenty-year rule here, but I have not yet read a comprehensive, compelling account of the factors that led to its dissolution--was it inevitable as early as 1922? In 1953? 1985? Certainly Western commentators had long discussed the eventual collapse of the USSR, but it didn't really seem imminent until Gorbachev, and even then the immediacy of the collapse wasn't recognized. I think we are probably a ways off from really looking through the Soviet Union of the 1980s and teasing out what was happening.

My question is, what's the process of writing history?

I am a bit skeptical of the term professional historians, as it is both too expansive and too restrictive. Amateur historians, those who lack affiliation and perhaps credentials, can still produce incisive, thorough, and compelling historical scholarship, while some professional historians may write shoddy papers or no papers at all -but perhaps they are phenomenal teachers!

The process of writing history varies from scholar to scholar, but in general you may start with a question ("why did the USSR collapse?"), read through the secondary literature (what other historians have already written), and then work through your primary sources. Maybe you're reconsidering how someone else interpreted historical actors ("was Gorbachev's anti-alcohol campaign effective?" might mean 'did it decrease public drunkenness, alcoholism, and mortality,' or it might mean 'did it change public opinions regarding alcohol consumption,' or 'did it precipitate or exacerbate negative/positive public perception of Gorbachev's policies'). After summing up the evidence--looking at newspapers, diaries, novels, advertisements, government records, and data on mortality rates, imprisonment, fines, alcohol sales, and so on, you interpret the evidence. What story does it tell? Maybe mortality rates go down (so, effective), but public perception also goes down. What does this imply? What if, after 1990, mortality rates skyrocket? 1991? History is complex and there is very rarely one right answer, but from this evidence you can sculpt a compelling story that meshes with the historical record.

Who writes them into history books that students read?

Anyone who wants to can write a historical monograph! Many books are published by historians employed as professors on the tenure track, but these scholars represent a minority of PhD graduates, in turn reflecting a minority of history undergraduates. More and more books are published by 'independent scholars,' oftentimes PhD graduates who are not employed in an academic setting. Textbooks can be written by these scholars as well as specialists in curriculum development, especially at the secondary school level.

How are these people selected and how do these people select what to write?

Tenure-track professors are PhD graduates who are selected by committees composed of fellow faculty members at their institution; this is a very competitive process, and it is not unusual for a single position to receives dozens or even hundreds of qualified applications. However, faculty members can (nominally) study whatever they wish, and in many cases their employment is contingent on completing a book -- oftentimes this is an edited version of the PhD dissertation.

Independent scholars receive no institutional support, but have wide latitude in what to write; the primary restriction, I imagine, is on what the scholar knows, as well as what resources they have access to.