The new atheist movement was a response to two things.
One, the 9/11 attacks, which demonstrated just how severe a threat suicide terror attacks facilitated by religious conviction could be. And...
Two, the rise of fundamentalist Christianity as a political force in the United States and the UK. Before the 9/11 attacks George W. Bush announced he would stop enforcing the rule that tax-exempt churches were not allowed to endorse specific candidates or referendums, and Protestant Evangelical churches immediately started preaching how to vote. (I don't know the specific mechanisms of religion in British politics, but it was a concern expressed by Dawkins as early as his 2002 TED talk.)
Dawkins was also specifically frustrated at (rather novice) attacks on biological evolution from Christian and Islamic organizations, who sought to change how science was taught in schools to adhere to their respective dogmas.
The new atheist movement was facilitated by the early internet, largely to raise awareness of specious religious arguments (against science, against LGBT+, against abortion access, in favor of corporal punishment, and so on) and also to assist nontheists, agnostics and philosophers to better define what they do or don't believe.
Also, atheists were (and still are) distrusted, more so than rival religions. Anti-atheist propaganda by religion-affiliated political organizations was sometimes put up on billboards and in schools. One of the missions of the new atheist movement is to counter prejudice against atheists that continues on to this day.
Since then, the new atheist movement has also a subset that helps religious victims of abuse deconstruct their faith. It turns out we have a whole lot of those coming out of Christianity and Islam, alike, and that seems to be the movement with the longest tail, and has only grown since major figures of the NAM have revealed their less-than-savory natures.
(Dawkins' arguments are robust, regarding atheism and addressing the religious debate sector. He just is an extreme dick, and has revealed himself to be so on social media.)