It is an indirect tax to the value added in the chain of production/selling of a product, being paid by the final consumer and collected from every previous member from the chain.
You're a final consumer when you cannot justify the buying as part of your "production process", thus you cannot claim a refund for the VAT.
For the ones in the middle of this chain, they add the VAT percetage to the product price (let say, 10%) to sell it, and collect the tax, while paying for the product or its part plus the same percentage over the price (since you earn money in this process and add value to the product, you're paying VAT over your costs that should be less than your income). In a monthly basis you net the bills payed and the bills collected, along with their respective VATs. The difference between the VAT you collected and the ones you paid would be the real tax ammount you have to pay.
As to prove it, you have to keep the copies from your bills and the bills you've paid. But you want to minimize your taxes, so you'll need a bill for every buy you make. The more bills you have, the less tax you'll pay. Thus, this enforces you to request the bills from your providers, since if you don't, you'd be paying the whole VAT tax from your pocket as if you were a final consumer. As a consecuence, every one in the chain maximize their income by avoiding the black market.
You may want to avoid collect the VAT and hide the transaction (i.e., sell in the black market), but your customer (the next guy in the chain) won't make the transaction unless you produce a bill allowing him to deduct the VAT.
For the IRS, the whole system can be controlled and the transactions audited: the copies from the bills you've made must match your customer's bill, both you and him reported them to the IRS as to minimize the tax, and you have to produce the bills and the copies under request.
So far, you have a virtuous cycle, the problem brings up with the final consumer, the one who ultimatelly pays the whole VAT from his pocket without a refund. He would request to buy from you without a bill to avoid the tax, but for you it'd be a problem to sell this way in a regular basis, because when you net the bills ti will look as you're losing money, which would be suspicious. Anyway, you could keep track fo your transactions as to sell to some of the final consumers without the VAT (let say, collecting half of the VAT amount for your pocket while saving the other half from your customer pocket), taking care not to look as if you were losing money. And if you manage to do so, this could kick back through the entire chain: you'd be able to make the same sort of arrangement with your providers, and so forth, eroding the system.
So the critical problem for the State is the way to control the final consumer's transactions.
There are other weak points. For example, in order to cheat, you may duplicate the bills, reporting and producing fake bills with lesser ammounts than the original ones (you reduce your claimed income, thus reducing the tax you'd pay from your pocket). This turns into a huge bureaucratic effort from the State in order to match the bills to detect the fraudulent ones, and in order to minimize this effort, this auditing process could be distributed by means of including new actors into the scene, like requesting the bills forms to be printed and enumerated by a trusted printing company whose identification will be stamped in every form, that reports this process back to the IRS.
Those problems have no easy solution, and the effort to solve them increases the costs, as an expense to the chain members, and indirectly as an operative expense to the State that must be covered by the same VAT. This increases the impact to the final consumer, reducing his aquisitive power and eroding the market. Because of this it is said the VAT is a regressive tax.