I'm not particularly familiar with SignalR, but here's what it looks like is happening, working from the inside out
String.Format("{0}: {1}" + Environment.NewLine, name, message)
Substitutes variables into a string. name would take the place of {0} and message takes the place of {1}. This would be the same as new String(name + ": " + message + Environment.NewLine)
RichTextBoxConsole.AppendText(String.Format(...))
Appends the formatted string into a RichTextBox
(Action)(() => RichTextBoxConsole.AppendText(...))
Declares a lambda expression that takes no parameters and calls the AppendText function.
(name, message) => this.Invoke((Action)(...))
Declares a second lambda function, taking 2 parameters (name and message) and calling the Invoke function with the previous lambda expression. I presume Invoke will actually execute the lambda expression.
HubProxy.On<string, string>("AddMessage", (name, message) => ...);
Again, not familiar with SignalR, but my educated guess would be that this is attaching the previous lambda expression to an AddMessage event, such that whenever AddMessage occurs, your lambda function will execute, causing the name and message of the AddMessage event to be appended into the RichText box via the Invoke function (perhaps to ensure the UI element is modified on the appropriate thread?)
Anyway, hopefully this helps you parse the line. Can't help you with converting it though >.<
(If I have said something incorrectly, please let me know and I will remove or adjust the offending statements)