What is the double bracket notation used here? Its kind of a bracket but I'm not sure what it means.

I have two ideas about it: It means the number of times the expression in satisfied or it changes for $1$ or $0$ depending on the result every time the $i$ value changes.
 A: This type of square-bracket is used in different context. One application is indeed for some sign functions.
In your example it is used identical to the so called Iverson Bracket. In this case the specific use of square brackets was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions.
But beyond your example beware, there are other applications such as $[[z]]$ could mean round down to the greatest integer less than or equal to $z$ etc.
A: In this instance, the symbol is an Iverson bracket, defined by
$$[\phi] = \begin{cases} 0 & \text{if}\ \phi\ \text{is false} \\ 1 & \text{if}\ \phi\ \text{is true} \end{cases}$$
where $\phi$ is some formula.
A: This is somewhat off-topic, but might help.  
Coxeter and Johnson, use square brackets to mark of a symmetry, eg [3,3] is the symmetry of {3,3}, but use double-square brackets to add a secondary extension ie [[3,3]] is the tetrahedral symmetry, along with swapping a figure and its dual.  For this group, central inversion works, but for [[3,4,3]] in 4D, one needs one of Conway/Thurston's 'wanders' to move the two.
A: It could very well just be a parenthetical statement (as in the English grammatical construct); e.g. that the equation to the left is true only if the condition on the right holds, or as an explanation as to why the equation is true, or maybe something else. This interpretation is encouraged by the appearance of what appears to be a period at the end of the equation on the left.
As others have said, actually showing the context would help clear things up.
