Does GABA help or hinder anxiety?
I wouldn't bother trying to interpret press-release science that doesn't even contain a citation or link to the original research. However, I will try to address some of the other content of your question.

It is indeed true that many anti-anxiety medications, particularly the benzodiazepines, act primarily to increase GABAergic activity by potentiating GABA receptors.

However, it is also true that GABAergic activity is not some monolithic attribute of the brain. Different brain regions have different functions, and some brain regions are associated with behavioral or neurochemical inhibition of other regions. An increase in GABA activity in a brain region that tends to inhibit others could theoretically be anxiety-inducing rather than anxiety-calming, even if the overall effect of increasing GABAergic effects is anxiety-calming. Looking at GABA levels is also only one part of the picture, and it fails to tell you a) how often GABAergic cells are releasing neurotransmitter, and b) how sensitive post-synaptic cells are to GABA. The overall effect of GABA is going to be roughly proportional to (GABA concentration) * (how often GABA is released) * (sensitivity to GABA).

An extreme example of why this is important are the delirium tremens that occur if a severe alcoholic suddenly stops drinking alcohol. Alcohol also potentiates GABA receptors, and with long term use the brain compensates by reducing GABA production. However, this adaptation is fairly slow, and alcoholics who quickly stop drinking can have motor symptoms and seizures as a result (some other neurotransmitter systems are involved as well, though, so this is a bit of a simplification). The opposite could be true as well: it's possible that increased GABA concentrations could be because not enough GABA is being released or because postsynaptic receptors have been downregulated. It is quite possible that you could have more GABA but less GABAergic inhibition. This study didn't test levels of inhibition, just levels of GABA.

Lastly, eating GABA to make you sleep is a bit silly and is far removed from brain levels of GABA or inhibition in the brain - I am not aware of any study that shows that eating GABA makes you sleepy; companies can freely put it in a drink and make abstract claims that suggest it aids sleep, but that doesn't mean it actually does, a beverage like that isn't regulated the same way that pharmaceutical drugs are.

In summary: be very wary of attributing a complex behavioral outcome to one particular element in the brain.