1. Ensure that your webgrid is wrapped inside a form.
@using (Html.BeginForm("Assign","Home"))
{
}
In the webgrid columns add a column with check box as follows which I think already you have.
grid.Column(header: "Assign?", format: @<text><input class="check-box" id="assignChkBx"name="assignChkBx" type="checkbox" value="@item.Id"/></text>)
2. At Controller Retrieve objects that are checked using id values as follows::
[HttpPost]
public ActionResult Assign(FormCollection form)
{
var ch = form.GetValues("assignChkBx");
foreach (var id in ch)
{
}
return View();
}