This is a phrase that means something like "a lot", "unbelievable", "amazing", or "horrible". Use it in casual speech when you're pretty excited or annoyed by something:
I have a pile of backed-up paperwork to take care of like you wouldn't believe!
This means "I have a huge pile of paperwork."
She's got artistic talent like you wouldn't believe.
This means "She's amazingly talented".
You can use "like you wouldn't believe" after a noun, as in the two examples given above. And you can also use it after a verb like this:
The soil here is great. The tomatoes have been growing like you wouldn't believe.
You should use "like you wouldn't believe" in casual spoken English, when you feel really energetic and you want to not only communicate your message, but also entertain your listeners. For example, it's good to use when you're telling a story.