You can download RTMP streams with freeware RTMPexplorer + rtmpsrv, CooJah + Blader, Orbit Downloader, StreamTransport, shareware Replay Media Catcher, WM Recorder...

And you can to capture RTMP streams in real time while you watch them using freeware RTMPexplorer + rtmpsuck, CooJah (without Blader), Orbit Grab Pro, shareware Replay Media Catcher (in the RTMP recording mode), WM Recorder (in the Loopback mode). Capturing can be done in real time only, but allows you to stay undetected.

If you use Linux, try rtmpsrv or rtmpsuck.

Related:
FAQ: How to record/download/capture/save Adobe Flash .FLV video RTMP stream from embedded player
How to record rtmp:// flash video .flv stream from embedded Adobe Flash Player