Like you, I used to think anime was pretty lame. As recent as a couple of years ago in fact. I always just dismissed it as being childish cartoons that Westerners who obsess over Japanese culture claimed to enjoy just because was Japanese. I wasn't a fan of the art style either. Big stupid looking eyes and crazy hair styles that nobody could ever pull off.
Then one of my friends told me they were watching an anime series. I think it was Cowboy Bebop and he'd also mentioned watching Fullmetal Alchemist: Brotherhood. I was pretty bored that summer and didn't have much to do so I decided to try FMA: Brotherhood for myself and it was actually pretty interesting to watch. Everything I used to hate about anime no longer seemed to bother me. I think I watched about 39 episodes before I eventually stopped watching. It wasn't until this week that I actually decided to start watching it again and I randomly got interested in anime. I've been watching various episodes of several series to find things I like and I think watching anime is going to be a new pastime of mine.
Why should you get into anime? Nobody's forcing you to. Watch a couple episodes of a series that sounds interesting to you if you're curious and then make your decision from there. Anime seems pretty varied in terms of themes and stuff.