Reactions happen when particles collide with enough energy. Anything that makes collisions more frequent or more energetic speeds them up.
Reading the graph. A steeper line means a faster rate. The line goes flat when a reactant is used up and the reaction stops. Speeding up a reaction makes the line steeper, but the final amount of product depends only on how much reactant you started with, so both lines finish at the same height. You measure the rate from the gradient (amount of product ÷ time).