Filling out a form like this is a start (when they're looking for dancers):
http://www.castingdancemoms.com/← Old page from last season
In the summer of 2015, casting is looking for dancers 6 to 12 years old; available for 16 weeks in the next 6 months; and with videos of them dancing online (on YouTube or Vimeo.) There's also a related open call audition in September, in Los Angeles.
Among other things, the producers want a link to video(s) of the dancer performing; and they want their mom to explain how they'd create fights and drama on the show...
(Submitting an entry for ALDC Dance Off might be an alternative to that initial first step. As mentioned below, applying for a related show, like AUDC, might also be a first step.)
Eventually, you'd have to pass an audition (or multiple types of auditions, likely including for the mom) with Abby and/or the producers.
A lot of people added to the show were previously on Abby's Ultimate Dance Competition. Asia, JoJo, Kalani... two of the new CADC dancers... here's some more. (I believe there were at least several more who first applied to AUDC, but who didn't get selected for that other show, for various reasons, including no production of a third season.)
A lot of people added to the show were previously attending ALDC Bootie Camps, either in Pittsburgh or L.A.
Several people had served as "ringers" dancing for the ALDC's senior team (that includes one of the most recent additions to Candy Apple's.)
In summary, I think it's about catching the attention of the producers in some manner, and letting them know you're willing and available. It can be helpful to to be "friends" with the right people here (including Abby)...
After catching the producers' attention, then it's mostly about convincing them that a particular dancer (and maybe their mom) will be an asset to the show, if they put them on television - and more of an asset than other people who'd like to be on the show.