You don't say what you're using to process the form, but you won't be able to process it with just HTML. So you'll need some sort of processing with another language.
As others have mentioned, you could do this with JavaScript. AJAX would be one way, but you could write your own code to do this if you really want to (but I wouldn't recommend it if you don't have to).
Another way to do this is with PHP. Have the page process itself when the form is submitted.
Similar to the PHP suggestion, you could do this via a CGI script/program. The action specified within the form would be to call the page itself. Here is a simple Python example, assuming the name of your script is so_self_call.py:
#!/usr/bin/env python
import cgi
import sys
def end_page():
print "</body>"
print "</html>"
def print_form():
print "<form action=\"so_self_call.py\">"
print "<input type=\"text\" name=\"theText\">"
print "<input type=\"submit\">"
print "</form>"
return
def start_page():
print "Content-Type: text/html"
print
print "<html>"
print "<head>"
print "<title>Example of Python CGI script which calls itself</title>"
print "</head>"
print "<body>"
return
def main():
start_page()
print_form()
the_form = cgi.FieldStorage()
if "theText" in the_form:
print "From last time, theText = %s" % (the_form["theText"].value)
end_page()
return
if __name__ == "__main__":
sys.exit(main())