Consider the below determinant $\triangle$:
$$ \triangle = \begin{vmatrix} a^3&a^2&a&1\\ b^3&b^2&b&1\\ c^3&c^2&c&1\\ d^3&d^2&d&1\\ \end{vmatrix} $$
"$\triangle$ is of the sixth degree in a,b,c and d" - This is the statement given about the above determinant.
What does "degree" of a determinant mean? and How determinant $\triangle$ is of sixth degree?
It is its degree as a polynomial in the a,b,c and d variables. In order to count this degree easier, you can look at
$$ \triangle (x)= \begin{vmatrix} (ax)^3&(ax)^2&ax&1\\ (bx)^3&(bx)^2&bx&1\\ (cx)^3&(cx)^2&cx&1\\ (dx)^3&(dx)^2&dx&1\\ \end{vmatrix} $$
Now you can think about polynomials only in the variable x rather than in 4 variables. If you compute the determinant, you will see that every summand is proportional to $x^6$. That is why you have of 6th degree in those 4 variables.