Why does a dominant allele mean a higher probability that offspring are affected by a disease?

To start with lets define an allele - these are different versions of the same gene. Genes are small sections of DNA which carry codes to tell your body what proteins to make. Each gene has two alleles - for example one could tell your body to make blue eyes and one brown eyes. Now the important thing here is that usually only one of these is expressed - and that will be the dominant allele. In this case the brown eye allele is dominant and that means it only needs one copy of the allele to give you brown eyes (if you have two copies you'll still have brown eyes)The blue allele in this case is recessive - this means that you need two copies of the blue allele to give you blue eyes.
In the case of diseases this means that for a dominant allele the parents only need to pass on ONE allele for the child to have this disease - giving them a 1/2 chance.In the case of a recessive disease the parents need to give TWO of the same disease allele (one from each parent) for the child to have this disease - resulting in a 1/4 chance.
We'll also cover punnet squares to make the chances easier to work out.