The details of the users are loaded by an Ajax call as you can see if you open the Developer Tools of your webbrowser:

You need to fed the DownloadString
method with that XHR url:
using(var wc = new WebClient())
{
wc.Encoding = Encoding.UTF8;
var all = wc.DownloadString("https://tauriwow.com/allrealm/ajax");
Console.WriteLine(all);
}
The content you get is html, so you have to do the parsing of that by yourself, maybe using a library like CsQuery
When you run above snippet this is what you get:
<div class="lft realmBox realmBoxFirsts">
<div class="realmTitle">
<div class="subtitle"><div class="lft"><b>»</b></div><div class="lft subtitle2">Tauri WoW Server</div><div class="rgt realmType">RPPvP</div></div>
</div>
<div class="realmStats"><a href="/allrealm/12">Detailed statistics</a></div>
<div class="realmDetails">
1) Acélember (<font style="color: lightblue;">20</font>)<br />2) Adjádékát (<font style="color: lightblue;">1</font>)<br />3) Adoresta