UPDATE: If you're following the bleeding-edge development versions, there's recently been added a much easier way to get a full-width message area for copying purposes, called the "bare display" mode!
It's bound to
meta-l by default—do
/key missing to set this keybinding if it's not already in your configuration—and it toggles off all of the WeeChat chrome, leaving just a log-style display of the current buffer's content.
If you don't have the development versions, the following solution using a bunch of settings still works, but I strongly recommend using bare display mode if you have it.
Well, you can fairly easily move the nicklist out of the way or hide it entirely. The commands for each are (respectively):
/set weechat.bar.nicklist.position top
/bar toggle nicklist
If you need the nicklist out of the way regularly, a keybinding may be added, like so.
/key bind meta-f /bar toggle nicklist
The left-side formatting is a little less straightforward. It can be disabled by making the following settings; this will eliminate the border for wrapped lines (so you don't get extra border characters) and disable the alignment of message sources.
/set weechat.look.prefix_align none
/set weechat.look.align_end_of_lines time
To re-enable, the defaults are:
/set weechat.look.prefix_align right
/set weechat.look.align_end_of_lines message
With all that done, you will likely still end up with hard wraps in your text! Setting the following option corrects this, but it is known for producing rendering bugs in WeeChat so it's not advisable to leave it on all the time:
/set weechat.look.eat_newline_glitch on
With all that set, copying from WeeChat directly should give you something pretty close to your log format.