Ok. So, whats the problem?
In button click event on server side event:
myTextbox.Text = "my dynamic text";
UPDATE:
Based on
wait are you asking that some one enters text into the text box, then clicks the button then the value is displayed on the webpage and if they do it again the new value and the previous value(s) are displayed?
This means you want to preserve the earlier text too. Ok.
Codebehind MyPage.aspx.cs:
allValuesOfTextbox += myTextbox.Text ;
someLabelOnWebpage = allValuesOfTextbox;
allValuesOfTextbox.Value += myTextbox.Text ;
someLabelOnWebpage = allValuesOfTextbox.Value;