God objects are a tempting consequence of bad software design and also badly implemented object orientation.
Essentially, a God object is an object with either too many methods or too many properties; essentially, it's a class that knows too much or does too much. The God object soon becomes tightly coupled to (referenced by) lots of other bits of code in the application.
So what's actually wrong with this? Well, in short, when you have one bit of code tied into every single other bit of code, you quickly find a maintenance disaster. If you adjust the logic for a method in a God object for one use case, you might find it having unintended consequences for another element.
In computer science, it is often a good idea to adopt a divide and conquer strategy. Often, big problems are just a set of little problems. By solving this set of little problems you can rapidly solve the overall problem. Objects should typically be self-contained; they should only know problems about themselves...