Holding our breath for a little while will have no effect on our breathing rate. However, when we hold our breath for a long time, the cells become oxygen starved and hence, when we start breathing again, the body rapidly tries to make up for this deficiency and the heart pumps faster to bring in oxygen at a faster rate than under normal, regular breathing session. This is what we experience as an increased breathing rate. When we breathe in, the oxygen enters our body and thus our blood; and when we breathe out, the carbon dioxide is released. However, when we hold the breathe, the carbon dioxide is no longer removed from the body and its level in our body starts increasing. The increased heart rate after we start breathing again will lead to drop in CO2 level.
So, basically the body becomes oxygen deprived when we hold our breath and it tries to make up for this decrease by faster breathing rate, when we start breathing again.