what exactly the assay of a chemical mean?

it's the amount of the actual chemical within the container.  if you have NaCl with an assay of 98% that means when it was assayed in quality control they found out that it was 98% NaCl.  The other 2% can be lost to impurities such as K.

It tells you how pure the compound is and what types of impurities are in it and gives you a percentage, or concentration of each impurity that you can expect to find in the sample. Generally the less impurities the more expensive the reagent.

The information in the assay helps you decide if the reagent is suitable for what you want to use it for.

It becomes very important for experiments that require very specific conditions. For example, in normal phase HPLC it is very important to have no water in your solvent, but many solvents contain small amounts of water, so when you go to buy HPLC solvents you need to buy high purity ones that have no traces of water.