Basically, the RV takes some values. Suppose you have X following a Bernoulli distribution (the most simple) with parameter p.
Then P(X=0)=1-p and P(X=1)=p.
What is the distribution of Y=aX+b ? Its only possible values will depend on those of X : a*0+b=b and a*1+b=a+b.
So P(Y=b)=P(a*X+b=b)=P(X=0)=1-p and P(Y=a+b)=p.