This quote is neither true nor false. The author called it a "sentiment", but I think when Kronecker made this statement, he took it as some kind of principle to work out the foundation of modern mathematics.

Basically the mathematical notion of integer is trying to make sense of our idea of a discrete unit. Taking a bite of apple makes it non-unit, but I think you would agree that the notion of "one apple" or "two dogs" makes common sense.

Also when reading someone's quote, I would always consider its context. I think Kronecket had the axiomatic construction of real number in mind. The idea of rational number and real number can be investigated by just assuming the idea of increment by one is solid.

These mathematical concepts are usually covered in a undergraduate analysis class. For more information, see Cauchy's construction and Dadekind's construction.