Q:

How does the proxy mechanism work with proxy settings in browser

We often find columns like Address, Port in web browser proxy settings. I know when we use proxy to visit a page, the web browser request the web page from the proxy server, but what I want to know is how the whole mechanism works? I have observed that many ISP allow only access to a single IP(of their website) after we exhausted our free data usage. But when we enter the site which we wants to browse in proxy URL and then type in the allowed IP, the site get loaded. How this works?

A:

In general, your browser simply connects to the proxy address & port instead of whatever IP address the DNS name resolved to.  It then makes the web request as per normal.
The web proxy reads the headers, uses the "Host" header of HTTP/1.1 to determine where the request is supposed to go, and then makes that request itself relaying all remaining data in both directions.
Proxies will typically also do caching so if another person requests the same page from that proxy, it can just return the previous result.  (This is simplified -- caching is a complex topic.)
Since the proxy is in complete control of the connection, it can choose to route the request elsewhere, scrape request and reply data, inject other things (like ads), or block you altogether.  Use SSL to protect against this.
Some web proxies are "transparent".  They reside on a gateway through which all IP traffic must pass and use the machine's networking stack to redirect outgoing connections to port 80 to a local port instead.  It then behaves the same as though a proxy was defined in the browser.
Other proxies, like SOCKS, have a dedicated protocol that allows non-HTTP requests to be made as well.

