Consider shooting a ray at the ocean at an oblique angle from a satellite: it bounces off and scatters away from you. Hardly any of the energy scatters back towards you.
Now, put a ship there. The ray bounces off the surface of the ocean and scatters up into the side of the ship, and from geometry, it's going to bounce off the ship and come straight back towards its original source. You get tons of energy coming back at you.
A ship on the ocean is basically a dihedral corner reflector, which is a very good target for a radar.
> I'm having a hard time imagining a sufficiently high radar resolution for such a wide sensor swath width at such an extreme range. Is the idea that you locate it with the wide sensor swath and then get a detailed radar signature from a more precise sensor?
That's one approach, there are so-called "tip and cue" concepts that do exactly this: a lead satellite will operate in a wide swath mode to detect targets, and then feed them back to a chase satellite which is operating in a high resolution spotlight mode to collect detailed radar images of the target for classification and identification.
However, aircraft carriers are big, so I don't think you'd even need to do the followup spotlight mode for identification. As an example, RADARSAT-2 does 35 meter resolution at a 450 km swath for its ship detection mode. That's plenty to be able to detect and identify an aircraft carrier, and that's a 20 year old civilian mission with public documentation, not a cutting edge military surveillance system. There are concepts for multi-aperture systems that can hit resolutions of less than ten meters at 500 km swath width using digital beamforming, like Germany's HRWS concept.
tl;dr: Radar works very well for this.