The Echo is a special ability that some rare individuals possess, and an important part of Final Fantasy XIV's storyline. Those that have been 'gifted', as the members of the Path of the Twelve calls it, have the power to resonate with other people's souls - and see their past, albeit not able to interfere or change it, as well as being able to speak every language in the world. It has been reported that this power is awakening over the last ten years, with no signal of its existence before that time.
So if the Power of the Echo was given to us, there's 2 possible sources:
1.) Althyk (the surveyor of change and space and god of time).
We can go into people's pasts and literally 'survey' what happened. When we 'return' to the present we retain the knowledge of what we saw and heard.

2.) A Primal.
If The Twelve didn't give us the power and Power of the Echo and more people seem to have been given the power in the last 10 years, and 10 years ago the Primals were either freed or awaken by the Battle of Silvertear Skies. Ifrit, Garuda, Leviathan, and Titan (maybe even Ramuh to the Sylphs? I remember the Sylphs saying they did not wish to summon their Primal, I think it was a NPC in Gridania during one of the quests) all seemed to seek out certain groups, it's not that farfetched to think that a Primal would seek Eorzeans. Again I think this backs up Anonymoose's theory that Siren is our Primal.
Anyone who possess the Echo is branded as a beastman by the Garlean Empire, and therefore is considered an enemy of the empire.
If we're now Beastmen because The Echo, and Primals only seek Beastmen, then sure why not. Also the Garleans are an advanced race who may have discovered knowledge about Primals or The Echo from other lands and aren't just arbitrarily branding us as Beastmen.