Because, like *all* inheritable mutations, they first occurred *in* reproductive cells. This is such basic biology that the article didn't bother to spell it out -- it would be like an article on a new airplane design not bothering to mention that aircraft need an atmosphere to fly and don't work in a vacuum.
Nothing in the article suggests the strange scenario you described, involving all the cells in a body mutating simultaneously. That such mutations must take place in germ cell lines is a "given".
If I am missing something, please, explain it to me.
See above.
P.S. Unlike you, I have never read creationist literature.
Then it's odd that your argument is a direct repetition of one of their standard canards. If you haven't read creationist sources, you've picked up their arguments indirectly.
This is pure logical analysis and evolution has never added up in my book.
If it were "pure logical analysis", it would not contain such elementary fallacies. For a description of how new mutations enter the population and eventually lead to speciation without there ever being a point where an individual has a problem mating with the members of the population to which he/she belongs, see this.
You might also try this. Not as authoritative as Ichneumon, but it's readable.