Bosola has the Duchess eat the apricots to confirm his suspicions that she is pregnant. Her pregnancy is confirmed when she throws up and goes into labor. It's suggested that they should say the apricots are poisoned so no one will suspect that the Duchess is giving birth.
I think this scene reflects the time period in which it was written. Knowledge of pregnancy and how pregnancy affected women was not great, so it's assumed that the Duchess will throw up (morning sickness?) even though most women suffer from nausea during the first few months of pregnancy, not at the end. Also, Bosola sees the sickness of the Duchess as proof positive of pregnancy, when in fact, throwing up doesn't always mean you're pregnant. Some women don't even crave certain things when they're pregnant.
He wanted to confirm her pregnancy. As a pregnant woman, she is constantly hungry and craving for food. By eating the apricots, she satisfy her hunger but the ill-effect of the apricot is it induces labour. Hence after eating the apricots, the Duchess went into labour therefore confirming his suspicion.
It was an Elizabethan belief that apricots would induce labour. Therefore, when she becomes ill and falls into premature labour... it confirms Bosolas suspicions.
Tropical fruits like pineapple and kiwi are 'said' to induce labour. Perhaps this was true of apricots in the 16 century?