What is proxy Browser?
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.