To find out where the nearest incursion to you is, open your mission journal and go to the incursions tab. If you can't see any incursions in the global report after refreshing it several times, you may need to clear your cache. To do this, press the escape key to open up the game options, go into the "Reset Settings" tab and click the button next to the text "clear all cache files." This will restart your game and the incursions will be visible. Once you arrive at a constellation that's under siege, a new chat channel will automatically open to help coordinate resistance efforts in that constellation. Players who are thinking of getting into the incursions are urged to proceed with caution and use only ships they would be willing to lose, as these upgraded Sansha ships are deceptively powerful for their size classes. If you'd like to take part in the incursions with other Massively readers, join the Massively channel in-game for instructions on how to get involved with the official Massively Mob corporation.
To give some background to the events taking place right now all over EVE, CCP has released a new video trailer putting the Sansha threat in context. Skip past the cut to watch the video, embedded in HD.