Antistatic mat , how it works?
First, the mat (and the connection to mains earth!!) should have a resistance that is high enough to cause no (extra) harm to you when 'normal' Voltages (let's say up to mains level) are involved. Think Mega-Ohms, and preferably more than one resistor in series to counter single-point-failure issues.

As for the open/closed loop part: think of your body as one plate of a capacitor, the other plate being the earth (in the form of all earthed metal in your surroundings). Now when you touch your mat, there is a closed path. The part that you think of as open (you and the earth) is a capacitor.

Another way to look at it is that for static electricity (as opposed to the more-or-less continuously current carrying electronics we are used to) a closed path is not a requirement. An excess of charge tends to spread itself out and preferably find a place with a shortage of charge to combine with. This is a temporary phenomenon, which lasts until the charges are evened out.