Implicit or procedural learning refers to learning how to do things like tying your shoes: It requires considerable cognitive effort and attentive control of muscles to get right, but eventually, after you've done it many times, you don't think about it anymore.
Driving is another example: After sufficient experience, you drive without conscious effort, becoming consciously aware of things only when they exceed your implicit bounds ("That car is too close, what's that unexpected shape at the side of the road").
The basic gist of the article is that mindfulness (which essentially makes you both more aware of and detached from your conscious cognitive efforts) mixes poorly with learning tasks or activities that are intended to eventually be performed automagically.
I suppose, I speculate, that the reason for this is that implicit learning requires you be "all in" attending to the task, but mindfulness would have you at least partially attending to your attention, so your mind is in fact distracting itself.
I further speculate that this likely "weakens" the "signal" your brain is trying to send from the explicitly conscious regions of your brain that are attending to the new task to the implicit, subconscious regions that will eventually perform the task, once it has been well learned.
(Just for fun, try to describe to someone in step-by-step detail how you tie your shoes. Then try tying them while describing what you are doing. Report back on how many attempts it takes before you succeed.)