User Agent is an HTTP header (i.e. one of the strings your browser sends to the server) which identifies... well, user agent, i.e. which browser is used to access the page
i.e. when accessing google.com, my browser sends something like this:
Accept-Encoding gzip, deflate
User-Agent Mozilla/5.0 (X11; Ubuntu; Linux x86_64; rv:11.0) Gecko/20100101 Firefox/11.0
Some sites use User-Agent header for "browser sniffing", i.e. to check which browser is used and to provide different code to different browsers. This was more popular ~10 years ago when browsers capabilities varied wildly.
Installing a Chrome extension would allow you to change the string your browser uses to identify itself (btw, Opera has this built in). This may fool some site, but it does not change browser behaviour (i.e. it does not make Chrome to render pages as IE not does it add support for, say ActiveX components - which, in fact, may be the reason why that site requires IE6)
So, provided the extension you're going to install is safe itself, the level of browser security will stay just the same.
(as a web developer I have serious doubts that any web site which works in IE6 only is worth visiting...)