Instead of having the mechanical moving antenna to scan, there are lots of smaller transmitter/receiver modules. Which give faster scanning rates and a more reliable radar with less maintenance because there are no moving parts, and no space neccessary for it to move, it occupies less space. The faster scan means a lower probability of the signal being intercepted/detected. Since they are newer/better/faster radars, they also offer better range. Google for more
The APG-63(V2) also has AESA and has already been fitted to some F-15Cs. The V3 is based on the APG-79's AESA of the Super Hornet.
The old radar was already multi-mode by the way.
can F-15C carry out attack missions with smart weapons now?
Short answer: Does it need to? No.
Longer answer:
Well no, not without further upgrades. The fire control system, stores management and other software needs to be upgraded. Which probably means that you need to upgrade your computer to accomodate the additional software, and need to upgrade your displays. For smart weapons you need integration of a targetting pod and/or GPS... but I guess the F-15 could let strike eagles handle that and just be a bomb truck thanks to its data-link, but you don't need a new radar for that. Finally you need the proper launchers, pylons which are wired and compatible with those launchers, but I think it's not really a problem as the F-15 was originally made capable of dropping dumb bombs.
Anyway, you'll end up with a degraded F-15E with only 2 to max 6 hard points suitable for A-S weapons. Or well, more like a F-16.