A mechanistic study aims to uncover a mechanism. It can be a mechanism of the disease (exactly how does the disease do damage to the body), or a mechanism of a drug (exactly how does the drug prevent or repair the damage) that is being investigated.
Understanding mechanism, be it mechanism of disease or mechanism of the drug, is very important toward targeted improvements of treatments, as opposed to trying lots and lots of compounds, hoping that someday we'll find one that's better.
Non-mechanistic studies involving drugs aim to show that the drug provides a clinically relevant health benefit over competing treatments, or to determine the safe dosage of a drug. At that point, the mechanism of action of a drug is known to a reasonably large extent.
In short, there are many different things that need to be known for drugs to reach the market. Mechanism is one of them.