Both air and water move under the influence of pressure changes. The movement is from an area of excess pressure to deficit pressure. When a straw is placed into a drink, the portion of the straw not in the drink will have air in it. When this air is removed by bringing the air into the mouth, there is a deficit of air above the liquid as compared to the liquid that is outside the straw. Thus, the liquid will move up the straw to try to balance this disequilibrium. The diagram below shows the movement of the liquid in the straw from higher pressure (excess pressure) toward lower pressure (deficit pressure).

This same concept works with a mercury-in-glass barometer. There is a vacuum of air above the mercury in the tube and thus the mercury moves up in the tube to a height that balances the pressure inside and outside the tube. The tube has calibrated measurements for the air pressure. When the air pressure increases outside, the mercury rises higher in the tube. When the air pressure decreases outside, the mercury falls in the tube. Mercury is used due to its high density. A less dense liquid such as water will be able to rises many feet in the tube and this makes a water barometer not as practical. However, it makes a straw a very practical device to drink a soda through!