As was mentioned earlier in this thread, the text seemed to have been changed in the 2.5 update but it is not accurate. Here's the text from the old client:
"For most classes, this ability gives you an increased chance to perform a double attack with your primary hand, and at the third rank, your double attacks with the primary hand will no longer fail. For Beastlords, it instead grants the Beastlord a small chance to do an additional attack with their mainhand. This additional attack stacks with double attack."
Only monks and rangers innately get triple attack. As the text implies here, none of the language regarding no-fail double attacks applies here. The AA instead grants beastlords some Triple Attack skill based on the rank.
I'll update the text in the patcher to clarify this point a bit more.
As a side note, it's worth mentioning that "double attack" and "triple attack" aren't the most descriptive terms for what is actually happening. You always have your main hand swing, then if you have Double Attack skill you have a chance to make another attack with your main hand (for monks and rangers this is 100%), and if you have Triple Attack skill you get another chance to make yet another attack. As a beastlord it is possible to successfully swing your main hand, fail your Double Attack chance, and succeed at your Triple Attack chance if you have the Melee Mastery AA. This would end up looking like 2 attacks even though your Triple Attack was the one that succeeded.