Asked another way, why to atheists "evangelize"? Who or what compels them to do this? Why do atheists care about what others believe?

First, let's note that not all atheists are vocal about their position. Outside of simply being apathetic, many are "in the closet", unable or unwilling to even let their friends and family know about their atheism, for fear of social or professional repercussions. Vocal atheists are actually a minority.

Of those who are vocal, some "bash religion" for fun. There's trolls in every crowd.

Many others, however, do so because they care about their fellow humans, and humanity in general. They see religion, and even faith in general, as being destructive to humans and humanity. For example, some parents allow their children to die from preventable diseases, due to faith that God/prayer would heal their child[1].

Our brains have a capacity for reason - to assess facts of reality and make decisions. This process is derailed by faith-based thinking and superstition.

We care about our fellow humans dying needlessly, suffering needlessly, and otherwise would like to live in a society that's safe, progressive and happy - not just for us, but anyone who doesn't conform to the "correct" religion. We're confronting a major source of derision, division and damage in society - religious mentality.

We promote reason, evidence and skepticism as its replacement. If you actually ask these people, they'll often tell you that they're not trying to generate more atheists. That's not actually the goal. It may be the outcome, but the focus on evidence and reason, and getting more people to engage in that, is what they're after.

Beliefs don't exist in a vacuum. They inform every action we take, from whether to take our child to the hospital, to whether we vote for the rights-killing political candidate. Beliefs matter, and affect us all.