More substrate molecules means a collision between substrate and enzyme is more likely, so more active sites will be occupied and more enzyme-substrate complexes will be formed.
rate of reaction only increases until a 'saturation' point. After that, there are so many substrate molecules that the enzymes have about as much as they can cope with (all the active sites are full), and adding more makes no difference - the enzyme concentration becomes the limiting factor.
Substrate concentration decreases with time during a reaction (unless more substrate is added to the reaction mixture), so if no other variables are changed, the rate of reaction will decrease over time too. This makes the initial rate of reaction (the reaction rate right at the start of the reaction, close to time 0) the highest rate of reaction.
