Depending on how you opted to tax your LLC, there may be a tax advantage to getting the company's profit down to 0. (Getting the bank account down to zero sounds like a bizarre old-wives' tale)...
Here are the details:
An LLC can be taxed as a pass-through entity (the default) or, if you filed IRS Form 8832, like a corporation.
If the LLC is taxed like a corporation, any profits left in the company will be subject to corporate taxes. When you then go to pay yourself, you'll pay income taxes, as well: this is called double taxation.
To avoid double taxation, corporations (and LLCs taxed like corporations) often try to pay out any profits from the company by the end of the year, usually in the form of salaries and bonuses to the owners. These salaries and bonuses have to be "reasonable" in the eyes of the IRS (you can't give your janitor a $10,000,000 bonus) or they will be correctly seen as a way to avoid income tax.
There are other ways to reduce the profit in the company for a given tax year: accelerate certain payments that can be paid either in December or January, or defer certain payments that are due to you from customers.
Smart corporations start planning in September to try and minimize the amount of profit left in the company at year end.
If the LLC is a pass through entity, its profit is added to your personal income tax no matter what you do, and there is no double taxation issue, so I don't see why you would want to get the profits down to zero... it won't matter. And as to why you should get your bank account to zero, that just makes no sense to me; it sounds like a bit of an old wives' tale.