You're assigning a function to your unassigned_requests
instead of performing actual filtering.
I think you need this:
function filterRequest(requests){
return requests.filter(function(el){
return el.Assigned_To === '';
});
}
$.when(SharePointJSOMService.getRequests())
.done(function(jsonObject){
$scope.requests = jsonObject.d.results;
$scope.unassigned_requests = filterRequest($scope.requests);
//or $scope.unassigned_requests = $scope.requests.filter(function(el){
// return el.Assigned_To === '';
// });
$scope.$apply();
}).fail(function(err){
console.info(JSON.stringify(err));
});