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Technically, you don't need a framework. If you're making a really really simple site (think of the web back in 1992), you can just do it all with hard-coded HTML and some CSS.
And if you want to make a modern webapp, you don't actually need to use a framework for that, either.
You can instead choose to write all of the logic you need yourself, every time. You can write your own data-persistence/storage layer, or - if you're too busy - just write custom SQL for every single database access. You can write your own authentication and session handling layers. And your own template rending logic. And your own exception-handling logic. And your own security functions. And your own unit test framework to make sure it all works fine. And your own... [goes on for quite a long time]
Then again, if you do use a framework, you'll be able to benefit from the good, usually peer-reviewed and very well tested work of dozens if not hundreds of other developers, who may well be better than you. You'll get to build what you want rapidly, without having to spend time building or worrying too much about the infrastructure items listed above.
You can get more done in less time, and know that the framework code you're using or extending is very likely to be done better than you doing it all yourself.
And the cost of this? Investing some time learning the framework. But - as virtually every web dev out there will attest - it's definitely worth the time spent learning to get massive (really, massive) benefits from using whatever framework you choose.
I'm not sure there's a clear-cut definition of "framework". Sometimes a large set of libraries is called a framework, but I think the typical use of the word is closer to the definition aioobe brought.
This very nice article sums up the difference between just a set of libraries and a framework:
A framework can be defined as a set of libraries that say “Don’t call us, we’ll call you.”
How does a framework help you? Because instead of writing something from scratch, you basically just extend a given, working application. You get a lot of productivity this way - sometimes the resulting application can be far more elaborate than you could have done on your own in the same time frame - but you usually trade in a lot of flexibility.