By grounding I basically mean some theoretical knowledge of what meditation/mindfulness is about, whether that be through a teacher (like the vid you posted) or through literature. I come from a bit of a Buddhist perspective (bit reluctant to call myself a Buddhist), where meditation and mindfulness is central to the practice. There is an absolute wealth of information out there (public libraries and university libraries are a goldmine for me) and like anything you try to separate the useful from the not so useful.
I'll just give you an idea of some of the concepts. You basically learn about being present (which is exactly what anxiety isn't), you learn not to be too judgmental, loving-kindness (this concept helps to promote inner peace), mindfulness (Buddhism acknowledges that humans suffer and doesn't try to deny suffering), letting go, and various meditation techniques.
That's just the tip of the iceberg though, there really is a mountain of interesting ideas in the Buddhist literature.
Some books I'm currently reading include
A Path With Heart:Jack Kornfield (couldn't put this down, very easy to read)
Zen and the Mind: scientific approach to Zen practice (this is a great book for skeptics like myself, tells you all about why meditation works on a psychological and physiological level)
The Way of Zen: Alan Watts (not a bad primer for comparative religion between East and West)
Obviously it's not necessary to be a Buddhist or even follow any of this stuff to alleviate anxiety, but I chose to study it because the practice is well established and there is just so many fantastic teachings to discover.
