Your code should work fine.  If you are seeing the entire row of the excel file printed as a single string then my guess would be it is a problem with the content of the excel file itself in which case you would have to parse the string somehow.  Post a screen shot or your, even better, the excel file itself online somewhere so we can take a look.
In the meantime, check out this code which creates its own excel file first and then reopens and saves using your code:
public void ConsoleWriteTest()
{
    //Create some data
    var existingFile = new FileInfo(@"c:\temp\temp.xlsx");
    if (existingFile.Exists)
        existingFile.Delete();
    using (var pck = new ExcelPackage(existingFile))
    {
        var workbook = pck.Workbook;
        var worksheet = workbook.Worksheets.Add("newsheet");
        #region Data
        //The data
        worksheet.Cells["A1"].Value = "Col1";
        worksheet.Cells["A2"].Value = "sdf";
        worksheet.Cells["A3"].Value = "ghgh";
        worksheet.Cells["A4"].Value = "sdf";
        worksheet.Cells["A5"].Value = "wer";
        worksheet.Cells["B1"].Value = "Col2";
        worksheet.Cells["B2"].Value = "Group B";
        worksheet.Cells["B3"].Value = "Group A";
        worksheet.Cells["B4"].Value = "Group C";
        worksheet.Cells["B5"].Value = "Group A";
        worksheet.Cells["C1"].Value = "Col3";
        worksheet.Cells["C2"].Value = 634.5;
        worksheet.Cells["C3"].Value = 274.5;
        worksheet.Cells["C4"].Value = 453.5;
        worksheet.Cells["C5"].Value = 634.5;
        worksheet.Cells["D1"].Value = "Col4";
        worksheet.Cells["D2"].Value = 996440;
        worksheet.Cells["D3"].Value = 185780;
        worksheet.Cells["D4"].Value = 686468;
        worksheet.Cells["D5"].Value = 996440;
        #endregion
        pck.Save();
    }
    //Reopen the file
    using (var package = new ExcelPackage(existingFile))
    {
        ExcelWorkbook workBook = package.Workbook;
        if (workBook != null)
        {
            if (workBook.Worksheets.Count > 0)
            {
                ExcelWorksheet currentWorksheet = workBook.Worksheets.First();
                var lastrow = currentWorksheet.Dimension.End.Row;
                var lastcol = currentWorksheet.Dimension.End.Column;
                for (int i = 1; i <= lastrow; i++)
                {
                    for (int j = 1; j <= lastcol; j++)
                    {
                        object asd = new object();
                        asd = currentWorksheet.Cells[i, j].Value;
                        Console.WriteLine(asd);
                    }
                }
            }
        }
    }
}
EDIT 1: Row or column as a collection
//Get entire rows or columns as collection and then print by casting
using (var package = new ExcelPackage(existingFile))
{
    ExcelWorkbook workBook = package.Workbook;
    if (workBook != null)
    {
        if (workBook.Worksheets.Count <= 0) 
            return;
        ExcelWorksheet currentWorksheet = workBook.Worksheets.First();
        var lastrow = currentWorksheet.Dimension.End.Row;
        var lastcol = currentWorksheet.Dimension.End.Column;
        //get the row of column headers which are strings
        var asdrange = currentWorksheet.Cells[1, 1, 1, lastcol];
        Console.WriteLine("As cell objects");
        foreach (var cell in asdrange)
            Console.WriteLine(cell.Value);
        object asd = new object();
        asd = currentWorksheet.Cells[1, 1, 1, lastcol].Value;
        object[,] cellObjects = (object[,])asd;
        List<string> stringList = cellObjects.Cast<string>().ToList();
        Console.WriteLine(Environment.NewLine + "As casted to a List");
        Console.WriteLine(stringList[0]);
        Console.WriteLine(stringList[1]);
        Console.WriteLine(stringList[2]);
        Console.WriteLine(stringList[3]);
        //get the second row which is a mix of strings and double
        asdrange = currentWorksheet.Cells[2, 1, 2, lastcol];
        Console.WriteLine(Environment.NewLine + "As cell objects");
        foreach (var cell in asdrange)
            Console.WriteLine(cell.Value); 
        asd = currentWorksheet.Cells[2, 1, 2, lastcol].Value;
        cellObjects = (object[,])asd;
        List<object> objectList = cellObjects.Cast<object>().ToList();
        Console.WriteLine(Environment.NewLine + "As casted to a List");
        Console.WriteLine(objectList[0]);
        Console.WriteLine(objectList[1]);
        Console.WriteLine(objectList[2]);
        Console.WriteLine(objectList[3]);
        //Get Col3 which are doubles
        asdrange = currentWorksheet.Cells[2, 3, lastrow, 3];
        Console.WriteLine(Environment.NewLine + "As cell objects");
        foreach (var cell in asdrange)
            Console.WriteLine(cell.Value); 
        asd = currentWorksheet.Cells[2, 3, lastrow, 3].Value;
        cellObjects = (object[,])asd;
        List<double> doubleList = cellObjects.Cast<double>().ToList();
        Console.WriteLine(Environment.NewLine + "As casted to a List");
        Console.WriteLine(doubleList[0]);
        Console.WriteLine(doubleList[1]);
        Console.WriteLine(doubleList[2]);
        Console.WriteLine(doubleList[3]);
        Console.WriteLine(doubleList[4]);
        Console.WriteLine(doubleList[5]);
        Console.WriteLine(doubleList[6]);
        Console.WriteLine(doubleList[7]);
        Console.WriteLine(doubleList[8]);
    }
}