I think these lines are in your webservers's error log, not in your PHP's error log?
They messages are harmless, and mean exactly what they say, that the client closed the Keep-Alive connection.
Normally, when you visit a website, you need to fetch a whole bunch of resources, the page itself, all the images, CSS, Javascript, favicon, etc. For these forum's index page for example, a total of about 25 pages need to be fetched.
In a simple model, the client makes a total of 25 connection. This means the client and server have to establish a TCP connection 25 times. A relatively costly operation.
With a HTTP Keep-alive, the server won't close the connection immediately after the file is sent, but keeps the connection open for a
n amount of seconds so that it can be recycled for more files. No more TCP handshaking overhead, meaning a faster page load.
There are a few additional advantages like less resources required of the server.
You didn't mention which webserver you're using, you can try settings your Keep Alive timeout lower, this is a good idea anyhow IMHO,
details here. The default in Apache is 15 seconds, something like 2 or 3 seconds makes much more sense IMHO.
