1. Adding the finger displaces water. By Archimedes' Principle, there is an upthrust on the finger equal to the weight of water displaced. In other words, a new force has been applied to the system. By Newton's Third Law, an equal and opposite force must act. The upthrust acted from the water on the finger. This new reaction force acts force from the finger on the water. Water is not compressible so the force is transmitted to the bottom of the beaker and pushes it down onto the scales. By this logic, the increased force should be equal to the upthrust and thus to the weight of water displaced. The extra mass is 5.53 grams. Water has a density of 1 gram per cubic cm. This means that the displaced water would have a volume of 5.53 cubic centimetres. I don't think that my finger is quite that big but it's not far off.
2. The second solution is that deeper water exerts a bigger pressure at the bottom. Displacing the water makes it deeper.
The effect would not be noticeable in a beaker of air for two reasons. Air is much less dense than water so the weight if displaced air is tiny and thus might not register on the scales, Secondly, air is compressible so the reaction force would probably just push the air molecules closer together rather than push down on the bottom of the beaker.
