Free TX accerelator was made to include a TX with an appropriate fee (not a low fee) manually to the next block of the X mining pool in order to give people the priority they need instead of spammers in the bitcoin network. These tx accerelator just needs a human verification (e.g. captcha). There is also ViaBTC - a mining provider - who is handing out free accerelation when they buy its cloud mining hashpower.
But what's their incentive? If miners want to include free/low-fee transactions, why don't they do it automatically?Unless the idea is to avoid blockchain spammers by requiring a captcha.
Bitfury does, or did do it automatically. By requiring someone to go to a website, it's nearly free advertising for them, makes them look like the good guy and it prevents spammers from getting in their low fee spam easily. It also lets people who need a confirmation really urgently get priority over others, since those that might be just consolidating their inputs might not be in a rush. They also offer a paid acceleration service, and some users of the free service may opt to use the paid one so it's easier for them to get their transactions confirmed.
