Looking for LWP::Debug::conns (I have lwp 5.803), I see it appear in LWP::Protocol::http (commented out) and LWP::Protocol::http10 (often used). My little test shows no output

perl -MLWP::Debug=+conns -MLWP::Simple -e get(shift) http://perl.org
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After looking at the LWP documentation, I enable PERL_LWP_USE_HTTP_10, and try again, and again no output.

Then I remember LWP::Simple is simple (grrr), so I adapt your sample and try again (oy)

#!/usr/bin/perl -w
use strict;
use LWP;
use LWP::Debug qw(+conns);

local $ENV{PERL_LWP_USE_HTTP_10} = 1;
my $agent = LWP::UserAgent->new();
my $url = 'http://www.perlmonks.org';
my $response = $agent->get($url);
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and this time it works :)

I would(you should) ask the author/maintainer about this, as it looks like something he's overlooked.