As for why you should root, there are many things that can only be done once you've rooted. Some examples are take screenshots, tether without a tethering plan (assuming your carrier enforces one and doesnt check alternate ways), and best of all install custom ROMs. ROMs are basically complete firmware replacements that offer functionality or UIs not present in the stock firmware. If you do flash one of these, make sure you read enough that you know what your doing and DO NOT flash a GSM rom or you will brick.
There is no application thats directly comparable to Cydia, but Google doesnt care if you're rooted so you can find root-only apps in the standard Android Market. Alternatively, you can sideload them from a PC.