For every Boolean function except the one identically equal to zero we can say that:
$$f(x_1,x_2,...x_n)=\bigvee [f(\alpha_1,...\alpha_n)\land x_1^{\alpha_1}\land...x_n^{\alpha_n}] \\ f(x_1,x_2,...x_n)=\bigwedge [f(\alpha_1,...\alpha_n)\lor x_1^{\alpha_1}\lor...x_n^{\alpha_n}]$$
where $\alpha_i \in \{0,1\}$
The use of $\lor$ and $\land$ outside the brackets and $f(\alpha_1,...\alpha_n)$ confuse me. Is this just a way of saying that any Boolean function can be written in disjunctive normal form and conjunctive normal form?
This would make sense if the $\bigvee[\cdots]$ means to take the disjunction of one instance of the formula in brackets for each possible combination of values for the $\alpha_i$s.
In that case then, yes, it's just a way of saying that the function can be written in disjunctive and conjunctive normal forms. Why they exclude the always-0 function is a mystery.
But it is not really standard to omit the index range like that. And the notation also seems to depend on using $x^0$ and $x^1$ to denote $x$ and $\neg x$ (or vice versa), which is definitely not a standard notation.