Although, yes, in practice it doesn't usually matter what order you write a truth table, this author seems to tacitly be making the assumption that you must write a truth table in Lexicographical Order. I.e, flip the furthest bit first.
Thought of in a different way, if you interpret the entries as a binary number, they are written from smallest to largest. $000_2 = 0, ~001_2=1, ~010_2=2, ~011_2=3,~\dots$
By making sure such truth tables are written in a very specific order, the representation the author refers to is then well-defined and unique.
Thinking of a boolean function on three variables as a function from $\mathbb{F}_2^3\to \mathbb{F}_2$, their notation $F=\sum(1,2,4,7)$ is just saying that $F(0,0,0)=0,~F(0,0,1)=1,~F(0,1,0)=1,~F(0,1,1)=0,~F(1,0,0)=1,~F(1,0,1)=0,~F(1,1,0)=0,~F(1,1,1)=1$ by simply only listing the ones with an output of one in shorthand notation based on where they appeared on the list. ($F(1,1,1)$ for example being the seventh entry on the list, where we start counting from zero, hence the $7$ in $\sum(1,2,4,7)$)
Best Answer
What is under sigma notation is '1's and everything else is '0's
so take this table as the format and fill up it using '1's and '0's according to the sigma notation you got :) and simplifiy is using K-maps