Fertilisers contain minerals that plants require to grow; most of them are called NPK fertilisers, this means they contain nitrates, phosphates and potassium.
Nitrates are needed to make proteins- proteins are what plant cells are made of. If there is a lot of nitrate in the soil then plants have the ability to grow as much as they can.
Phosphates are involved in respiration and growth- both things are needed to sustain a plant.
Potassium must be present for enzymes to work- with out it the plant wouldn't be able to carry out reactions and so would die or have very limited growth.