Q:

what does "value" represent in d3 force directed graph

What does value do in the example of a force-directed graph made with d3?

A:

If you look at this block of code from the example you gave:
var link = svg.selectAll(".link")
    .data(graph.links)
    .enter().append("line")
    .attr("class", "link")
    .style("stroke-width", function (d) {
        return Math.sqrt(d.value);
    });

You can see that the value attribute in the JSON file is used to set the stroke-width of the line that connects that node to its target node.
Specifically this part of the function:
.style("stroke-width", function(d) { return Math.sqrt(d.value); });

