Understanding the conformal invariance of the scalar massless wave equation

conformal invariance means we have the all of the following:

1. rotational invariance
2. scale invariance
3. special conformal invariance
4. translation invariance

now since wave equation does not have any potential term it fullfills 4. since it is scalar and again does not have any external field being applied it full fills 1. now since it is massless it fullfills 2. mass $m$ has a dimension same as momentum it is one over length. thus because of that under scale transformation, the terms with $m$ in equation of motion changes. Since in your case there is no mass term it full fills 2. 3. is not very important 1 2 4 are important and this the intiuitive explanation.