Because they are kids with immature immunes systems and the ailments are common. If the immune system has not met the pathogen before, it does not respond quickly or in force until after the pathogen has caused symptoms. After that it retains a memory for it and will come down like the fist of God if it meets that pathogen again.
This is why common cold viruses mutates like SumBees trying to survive the human immune system.
As for kids getting sick after getting shots it's because they just visited a doctor's office, which is a cesspool of coughing children with uncovered mouths. I'm always surprised when the kids don't get sick after a check up.
As you point out the immature immune system is working to fight the new pathogen that was just placed into the body. Sometimes it takes time for caused symptoms. During this time the bodies immunity is weakened working on this pathogen. The person is now susceptible to common aliments.
This is the pattern for the kids that get autism. They get sick from common aliments after taking these vaccines.