All the organisms in a food web are related to one another and changes of population of one organism will affect the population of others. Each organism, in a food web, is either a prey or predator (or both) of another organism. Thus, changes in its population will mean less threat or food to another organism. Additionally, many organisms are in competition, with each other, for same food source. Thus, changes in the population of an organism will affect the organism that serves as its food or the organism that feeds on it and the organisms that compete with it.
For example, deer eat grass and are in turn, eaten by tigers. Deer also compete with other herbivores, such as say, cows, etc. If the population of deer is increased, cows will have more competition for food, more grass would be consumed and tigers will have more food. This could result in lesser grass and cow population and more tiger population.
If we completely remove deer, tigers will have less food, cows will have more food and grass will have lesser predators. Thus, grass density will increase, cows population will increase and tiger population will decrease (in short term; in longer term, additional cows will feed tigers).
Hope this helps.