<asp:Content id="Head" ContentPlaceHolderID="TemplateHead" runat="server">
    <title>Patient Safety Event Report</title>
   
    <script type="text/javascript" language="javascript">
           alert('recongized javascript');

           function rBtnClick() {
               alert('inside rbtnclick');
               var radio = document.getElementsByName('RadioButtonList8');
               alert('assigned radio');

               for (var i = 0; i < radio.length; i++) {
                   alert('inside for');
                   if (radio[i].checked) {
                       alert('a button has been checked');
                       if (radio[i].value == 'Yes') {
                           alert('displaying explanation box');
                           document.getElementById('question8Yes').style.display = 'block';
                       }
                       else {
                           alert('removing explanation box');
                           document.getElementById('question8Yes').style.display = 'none';
                       }
                   }

               }
              
           }

           function textBoxValidation(oSrc, args) {
               alert('inside textbox validation');
               var radioButtons = document.getElementsByName('RadioButtonList8');
               alert('assigned radiobutton list');
               var textBox = document.getElementById('txtQuestion8');
               alert('assigned text box');
               for (var x = 0; x < radioButtons.length; x++) {
                   alert('inside for loop');
                   if (radioButtons[x].checked) {
                       alert('inside if radiobuttons checked part');
                       if (radioButtons[x].value == 'Yes') {
                           alert('radiobutton yes was checked');
                           if (textBox.value.length < 1) {
                               args.IsValid = false;
                               alert('Yes checked textbox empty');
                           }
                           else {
                               args.IsValid = true;
                               alert('yes checked textbox has something');
                           }
                       }
                       else {
                           args.IsValid = true;
                           //alert('yes was not checked');
                       }
                   }
               }
           }

           alert('finished recognizing javascript');
   
    </script>

<asp:Content id="Main" runat="server" ContentPlaceHolderID="TemplateBody">
    <div id="q8">
        <b>8.  Was any....?</b>
        <asp:RadioButtonList ID="RadioButtonList8" runat="server" >
            <asp:ListItem Text="Yes" Value="Yes"></asp:ListItem>
            <asp:ListItem Text="No" Value="No"></asp:ListItem>
            <asp:ListItem Text="Unknown" Value="Unknown"></asp:ListItem>
        </asp:RadioButtonList>
        <asp:RequiredFieldValidator ID="RequiredFieldValidator8" runat="server" ControlToValidate="RadioButtonList8" Text="*"></asp:RequiredFieldValidator>
        <div id="question8Yes" style="display:none">
            If Yes, please specify:
            <br />
            <asp:TextBox ID="txtQuestion8" runat="server" MaxLength="250" Width="350px"></asp:TextBox>
            <asp:Label ID="Label1" runat="server" Text=""></asp:Label>
            <asp:CustomValidator ID="CustomValidator1" ControlToValidate="txtQuestion8" runat="server" ValidateEmptyText="true"
                ClientValidationFunction="textBoxValidation" ErrorMessage="Please expalin why" Text="*"></asp:CustomValidator>
        </div>
       
        <br />
    </div>

</asp:Content>

Since RadioButtonList8 is an ASP.Net server control you can't access directly using server ID because it creates different DOM IDs while rendering the page

 This is wrong

var radioButtons = document.getElementsByName('RadioButtonList8');

You would need to change your code like below

<script type="text/javascript">
        function rBtnClick() {
            document.getElementById('question8Yes').style.display = 'none';

            var radiobuttonList = document.getElementById('<%= RadioButtonList8.ClientID %>');
            var options = radiobuttonList.getElementsByTagName('input');
            for (var i = 0; i < options.length; i++) {
                if (options[i].value == "Yes" && options[i].checked)
                    document.getElementById('question8Yes').style.display = 'block';
            }
        }
    </script>

Balaji