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.

Let's say I connect myself to the mat using an anti static wrist strap, how is it supposed to help me discharge my static charge when there is no closed loop between me, the mat, and the ground?

As Wouter says, your body can behave like a capacitor, storing electric charge. A good example of a body storing charge is in the winter, removing a sweater or shuffling your feet across a carpet, then touching a doorknob - ZAP! That tiny blue spark is thousands of volts of static electricity jumping off your charged body. The antistatic mat is very slightly conductive. So if you put on the wrist strap or touch the mat, those thousands of volts are harmlessly coupled to ground. Current does flow - from your body, through the mat, and to ground - simply because your body has substantially more charge on it than ground does, which is an imbalance - and charge prefers to equalize.

And lets say I am using a power supply which I am feeding from the mains, and the positive clip of the supply touches the mat accidentally. Wouldn't it be a dangerous scenario for me to be in contact with the mat?

If the mat were very conductive, yes. But the mat isn't - it's only very slightly conductive - the equivalent to Megaohms of resistance. And it has to be, to dissipate those thousands of volts safely and slowly. The resistance is so large, that any type of "shorting out a benchtop power supply" would be imperceptible by us.