It looks to me like the key phrase are in the word "Peer" followed by, what, an "L" or an "R"? (sounds like "left" or "right" to me, judging by this short sample), the IP address, and the port number. Good, let's try to match that.

local $\ = "\n";  # line feed at end of print
while(<>) {
    if(/\bPeer: (\w):\s+(\d+\.\d+\.\d+\.\d+):\s*(\d+)/) {
        print $2;
    }
}
[download]

Does that help?

n.b. I've foreseen some requirement changes so I'm capturing the letter ("L", "R") and the port number too.

Update: it looks to me like you edited your question to include the time too. Fixing:

 
local($\, $,) = ("\n", " "); 
while(<>) {
    if(/([\d:.]+).*?\bPeer: (\w):\s+(\d+\.\d+\.\d+\.\d+):\s*(\d+)/) {
        print $1, $3;
    }
}
[download]