What it is: It's essentially a work-around to the once-a-day per toon World Quests, allowing you to complete Family Familiar at a much faster rate than you would normally.
Characters with World Quests unlocked will find that they are only able to battle the active Broken Isles Pet Battle trainers once a day. Winning the encounter will complete the quest, making it impossible to repeat that same battle on the same day with that toon. You have to wait for the quests to rotate and hope each day that one of the Family Familiar trainers pops up.
Using the alt character-strat bypasses this, and allows you to battle that same trainer repeatedly on the same day (only on that specific character). Since the alt doesn't really complete anything (other than the Pet Battle), it doesn't incur a lockout and you'll be able to start and finish the trainer battle over and over. You won't receive any rewards on that character of course, but progress for Family Familiar can be reward enough. :)
This may seem like an exploit, however it was noticed and brought up during Legion Beta. I asked a developer if this was intended behavior for Pet Battle World Quests, and it seems that it is intended. So unless this has changed since beta, this strategy for completing Family Familiar faster is not an exploit.
How to do it: The only requirement is to have a character at max level (110), and DO NOT unlock World Quests.
This means you want to avoid completing the 'earn friendly reputation with X Broken Isles factions' milestone. Alts that are not level 110 will not work for this.
Once you've reached max level, log onto your main character (one that has World Quests unlocked) and see where the Broken Isles Pet Battle quests are located for the day. You can then take that information and hop onto your second character, travel to that location, find the trainer, speak with them and then engage in a Pet Battle over and over.
Note that the Suramar Pet Battle trainers are not included in this. Because they are unlocked during the Suramar storyline, alts without access to World Quests will not have Suramar progress and therefore will be unable to battle these trainers. They can see them and interact with the NPCs, but they won't be able to challenge them to a Pet Battle.
And that's about it!
Last but not least, if you're wondering what addon allows me to track Family Familiar in such an easy and convenient manner in the screenshot above, it's Family Familiar Helper! Made by Gello, who also created and maintains the awesome addon Rematch. Both are must haves for any pet collector/battler.