How can I use async to increase WinForms performance?

Yes, you're still doing all the work on the UI thread. Using async isn't going to automatically offload the work onto different threads. You could do this though:

private async void button2_Click(object sender, EventArgs e)
{
    string file = files[0];
    Task<string> task = Task.Run(() => ProcessFile(file));       
    rtTextArea.Text = await task;
}

private string ProcessFile(string file)
{
    using (TesseractEngine tess = new TesseractEngine("tessdata", "dic", 
                                                      EngineMode.TesseractOnly))
    {
        Page p = tess.Process(Pix.LoadFromFile(file));
        return p.GetText();
    }
}

The use of Task.Run will mean that ProcessFile (the heavy piece of work) is executed on a different thread.

You can also do this by starting your task in new thread. Just use Thread.Start or Thread. ParameterizedThreadStart

See these for your reference:

http://msdn.microsoft.com/en-us/library/system.threading.parameterizedthreadstart.aspx

Start thread with parameters