Проблема в том, что вы пытаетесь удалить с помощью индекса, а не с помощью идентификатора. Внесены некоторые изменения в функцию addRec и строку в html, вызывающую ее. Пожалуйста, пройдите их один раз:
var app = angular.module('myApp',[]);
app.controller("homeCtrl", function($scope) {
$scope.items = [{
itemID: 'BR063',
itemValue: 'sagsfgjkfdsffsdfsd'
}, {
itemID: 'BR06417',
itemValue: '1231231231123'
}];
$scope.selectedItems = [];
$scope.addRec = function(result, i,itemId){
if(result == true){
$scope.selectedItems.push($scope.items[i-1]);
}
else{
var index=0;
var isMatched = false;
angular.forEach($scope.selectedItems,function(item) {
if(isMatched === false) {
if(item.itemID === itemId) {
isMatched = true;
}
else {
index++;
}
}
});
$scope.selectedItems.splice(index,1);
}
}
});
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/angularjs/1.6.9/angular.min.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/angularjs/1.6.9/angular-route.js"></script>
<div ng-app = 'myApp' ng-controller="homeCtrl">
<h1>Select Rows</h1>
<table style="width:50%" class="table-responsive table-bordered ">
<tr>
<th class="text-center">Index</th>
<th class="text-center">Item ID</th>
<th class="text-center">Item Values</th>
<th class="text-center">Select</th>
</tr>
<tr ng-repeat="x in items track by x.itemID">
<td class="text-center">{{$index+1}}</td>
<td class="text-center">{{x.itemID}}</td>
<td class="text-center">{{x.itemValue}}</td>
<td class="text-center"><input type="checkbox" ng-model="itsVal" ng-change = "addRec(itsVal, $index+1,x.itemID)";/></td>
</tr>
</table>
<h1>Selected Rows</h1>
<table style="width:50%" class="table-responsive table-bordered ">
<tr>
<th class="text-center">Index</th>
<th class="text-center">Item ID</th>
<th class="text-center">Item Values</th>
<th class="text-center">Select</th>
</tr>
<tr ng-repeat="x in selectedItems track by x.itemID">
<td class="text-center">{{$index+1}}</td>
<td class="text-center">{{x.itemID}}</td>
<td class="text-center">{{x.itemValue}}</td>
<td class="text-center"><input type="checkbox" /></td>
</tr>
</table>
<div>