With a 3-button mouse, Cygwin rxvt acts the way it would under X. Use mouse button 1 (usually the left mouse button) to mark text. Rxvt automatically puts the marked text in the copy/paste buffer. Click mouse button 2 (the middle mouse button) to paste text at the cursor position.
My laptop has only two mouse buttons. There is no middle button, so there is no obvious way to paste in rxvt. MikeyMcKay points out that you can use Shift Mouse 1 (Shift+LeftMouseButton) to paste into rxvt. Thanks, Mikey.
More authoritatively, from the rxvt man page:
TEXT SELECTION AND INSERTION
The behaviour of text selection and insertion mechanism is similar to
Left click at the beginning of the region, drag to the end of
the region and release; Right click to extend the marked region;
Left double-click to select a word; Left triple-click to select
the entire line.
Pressing and releasing the Middle mouse button (or Shift-Insert)
in an rxvt window causes the current text selection to be
inserted as if it had been typed on the keyboard.
As an added bonus, Shift-Insert works in Emacs, too.