Saving Eggplant Seeds:
Eggplant seeds should be started early indoors. I like to sow mine mid-March, the same time as peppers. They do best when provided lots of light and warmth, I've found using grow lights makes a huge difference getting them started.
Transplant outdoors after summer warmth has arrived, usually mid-June around here. Choose a site with maximum sun. Eggplants do best in fertile soil with plenty of compost, but don't over-do the nitrogen, which can lead to wild foliage at the expense of fruit. They don't require staking the same way tomatoes do, and are fairly low maintenance once they get going.
One thing you'll want to keep an eye out for though, is the Colorado Potato Beetle... that most dreaded of insect herbivore. They love eggplant leaves even more than potato. My tip: pick off the bright-orange egg clusters which get laid on the underside of the leaves. They're easier to spot than the larvae themselves.
For seed saving purposes, you'll want to let your fruit ripen fully, well beyond their peak eating stage. Fully mature eggplant will usually turn a shade of yellow or tan, and will be noticeably less firm than they once were. If in doubt, cut a fruit open and check out the seeds; they'll be solid and brown when mature.
The trick now is to separate all these seeds from the flesh. The best method I've found yet is a pretty simple one; just grate them with a cheese grater. The seeds should be hard enough to (mostly) slip right through the blades. Toss your grated seedy eggplant into a bucket when you're finished. The next step to rinse and decant the seeds. Fill your bucket with water, swishing the pulp with your hand as you go. The seeds should separate themselves from the pulp fairly easily, and sink to the bottom. Carefully pour off the pulp. Any floating seeds are likely to be low viability, so don't worry about losing a few. Repeat the process until you have nothing left but fairly clean seeds! Now spread them on a tray to dry, and you've got eggplant seeds.