I don't know what the summation in the above pic is saying.
Assume that the x's (1 to n) shown above are numbers that have already been assigned values. n is a fixed value.
(I found this in Atiyah-Macdonald in a proof of some proposition. I tried looking at alternate proofs of the proposition online seeking an explanation for what the summation notation meant but all the proofs were really different from the one in the book.)
Note, that at every summation step $i$, $x_i$ is missing in the product, so I'd suggest that, as the index counts upwards, the missed out value in the product moves through the $x_i's$, i.e.
$$\sum_{i=1}^nx_1x_2\dots x_{i-1}x_{i+1}\dots x_n= (x_2\dots x_n) + (x_1x_3\dots x_n)+\dots +(x_1\dots x_{n-1})$$
Or, in a little more compact notation:
$$\sum_{i=1}^nx_1x_2\dots x_{i-1}x_{i+1}\dots x_n=\sum_{i=1}^n\prod_{j\leq n, j\neq i}x_j=\prod_{j\leq n, j\neq 1}x_j+\dots +\prod_{j\leq n, j\neq n}x_j$$