The first thing to notice is that each term, after expanding the brackets, is a cubic unit, that is, the exponents added together sum to 3. That suggests factors a+b, c+d, e+f where the letters are variables, six in all. But the expression only contains three variables.
The second thing to notice is that if we expand (a+b)(c+d)(e+f) we get 8 terms. And if we expand the given expression by removing the brackets we get 7 terms. If we write 2xyz as xyz+xyz then it contains 8 terms. So we can match the 8 terms.
Expand the factored expression:
ace+ade+bce+bde+acf+adf+bcf+bdf.
Now reduce the 6 variables to just 3 by equating a=d, b=e, c=f, so d, e and f can be replaced (note that none of the replacements are in the same factor as what replaces them, that is a and d are in separate factors, as are e and f):
abc+a²b+b²c+ab²+ac²+a²c+bc²+abc=a²b+b²c+ab²+ac²+a²c+bc²+2abc
This is structurally identical to the given expression.
So we can now replace a, b, c with x, y, z.
Factorisation is (x+y)(z+x)(y+z).