I have the following:
public partial class Form1 : Form
{
    public Form1()
    {
        InitializeComponent();
    }
    private void button3_Click(object sender, EventArgs e)
    {
        try
        {
            var strExpression = @"
            import sys
            sys.stdout=my.write
            print 'ABC'
            ";
            var engine = Python.CreateEngine();
            var scope = engine.CreateScope();
            var sourceCode = engine.CreateScriptSourceFromString(strExpression);
            scope.SetVariable("my", this);
            var actual = sourceCode.Execute<string>(scope);
            textBox1.Text += actual;
        }
        catch (System.Exception ex)
        {
            MessageBox.Show(ex.ToString());
        }
    }
    public void write(string s)
    {
        textBox1.Text += s;
    }
}
But I am getting an Exception that says there is no write.
What am I doing incorrectly?
You can set a stream and a textwriter directly from c#:
engine.Runtime.IO.SetOutput(stream, txtWriter);
engine.Runtime.IO.SetErrorOutput(stream, txtWriter);
To redirect the output for example you could override TextWriter class with a new one writing on your textbox.
in my application I did an override of StreamWriter class that rises events when something is written on the stream (here just a part of the code):
public class MyEvtArgs<T> : EventArgs
{
    public T Value
    {
        get;
        private set;
    }
    public MyEvtArgs(T value)
    {
        this.Value = value;
    }
}
public class EventRaisingStreamWriter : StreamWriter
{
    #region Event
    public event EventHandler<MyEvtArgs<string>> StringWritten;
    #endregion
    #region CTOR
    public EventRaisingStreamWriter(Stream s):base(s)
    { }
    #endregion
    #region Private Methods
    private void LaunchEvent(string txtWritten)
    {
        if (StringWritten != null)
        {
            StringWritten(this, new MyEvtArgs<string>(txtWritten));
        }
    }
    #endregion
    #region Overrides
    public override void Write(string value)
    {
        base.Write(value);
        LaunchEvent(value);
    }
    public override void Write(bool value)
    {
        base.Write(value);
        LaunchEvent(value.ToString());
    }
    // here override all writing methods...
    #endregion
}
Then in your application you should just do something like:
    MemoryStream ms = new MemoryStream();
    EventRaisingStreamWriter outputWr = new EventRaisingStreamWriter(ms);
    outputWr.StringWritten += new EventHandler<MyEvtArgs<string>>(sWr_StringWritten);
    var engine = Python.CreateEngine();
    engine.Runtime.IO.SetOutput(ms, outputWr);
    engine.CreateScriptSourceFromString("print 'hello world!'").Execute();
    void sWr_StringWritten(object sender, MyEvtArgs<string> e)
    {
        textBox1.Text += e.Value;
    }
                        Your example is close to working.
The problem you saw is because sys.stdout=my.write should be sys.stdout=my.
It also appears that Python expects to find a boolean softspace attribute.
I have made these two changes in the code below. Hopefully this should now work as you expected.
public partial class Form1 : Form
{
    public Form1()
    {
        InitializeComponent();
    }
    private void button3_Click(object sender, EventArgs e)
    {
        try
        {
            var strExpression = @"
import sys
sys.stdout=my
print 'ABC' ";
            var engine = Python.CreateEngine();
            var scope = engine.CreateScope();
            var sourceCode = engine.CreateScriptSourceFromString(strExpression);
            scope.SetVariable("my", this);
            var actual = sourceCode.Execute(scope);
            textBox1.Text += actual;
        } catch (System.Exception ex) {
            MessageBox.Show(ex.ToString());
        }
    }
    public bool softspace;
    public void write(string s)
    {
        textBox1.Text += s;
    }
}
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