I think that you're confused about what happens on the client (within the browser) and what happens on the server (in your ASP.NET code).
Most events on server-side controls, like the ASP.NET button, are handled server-side, through a mechanism called postback. The generated code on the browser causes a form submit to the Web server, where the page and its controls are re-created. The form data is parsed and the button control generates (in your case) a Click event for your server-side code. This code regenerates a new HTML page which will replace the old one.
Fortunately for you in this case, the ASP.NET button has a property "OnClientClick" where you can insert JavaScript code to do whatever you want (like change the label text) directly in the browser.
See MSDN for the docs on the property and an example of using both Click and OnClientClick.
Note that this is not related to SharePoint, this is a pure ASP.NET issue.