An unhandled exception of type 'System.StackOverflowException.Cannot evaluate expression because the current thread is in a stack overflow state

Both your getter and your setter are calling themselves. Remember that properties are just groups of methods, effectively. Your class really looks like this, but with a bit of extra metadata:

public class ConnectionElement : ConfigurationElement
{
     public string GetUri()
     {
         return (string) GetUri();
     }

     public void SetUri(string value)
     {
         SetUri(value);
     }
}

Now can you see why it's failing?

It's not clear why you're casting in the getter, but it looks like an automatically implemented property might do:

[ConfigurationProperty("uri", DefaultValue = "/")]
public string Uri { get; set; }

EDIT: If, as noted in comments, you're really trying to use the indexer, you need to do that instead of calling your property recursively:

public class ConnectionElement : ConfigurationElement
{
     [ConfigurationProperty("uri", DefaultValue = "/")]
     public string Uri
     {
         get { return (string) this["uri"]; }
         set { this["uri"] = value; }
     }
}