The conspiracy theories surrounding the Rothschilds all derive from a mix of the usual anti-Semitism coupled with typical Eurocentrism, a total misunderstanding of international finance, and of course a plethora of "historical" evidence that couldn't stand up to a Wikipedia article in terms of rigour.

As one of several wealthy banking families in Europe, the Rothschilds became known for their lavish lifestyle and significant construction of palaces. It was only natural that the accumulation of such wealth and extravagance would breed allegations of sinister means, which was perhaps not helped by the fact that the Rothschilds were Jewish, a historically popular target for demagoguery in a Christian Europe.

The real reason that the Rothschilds were able to become so wealthy was due to their acumen in creating financial derivative products and supplying financial liquidity to the right targets. By this, I mean that they primarily loaned to governments on their terms. Governments required lots of money, especially in war time, and the most well-known branch of the Rothschilds-the British one-was known to be a primary financier for the British government during the Napoleonic wars. However, they were more than just a source of funds-while they certainly put in some of their own money, their real strength came from their investment banking business, which allowed them to reach out to a network of investors, thus giving them access to a large pool of money to make loans or otherwise invest into a particular business. Consider for instance Goldman Sachs today: while Goldman Sachs does have a direct investment business (their Private Equity group, for example) their primary business has always been investment banking-namely, the business of working to get investors to pool their money together to buy stock or bonds of a company or other entity. This is precisely what the Rothschilds did and continue to do today.

The fact that their investments paid off-and that they were able to find parties desperate enough to pay them high levels of interest-gave them much increased financial strength and prestige. They became a primary source of funding for not only the British government but also the Prussian and even the Japanese, who were looking for financing to sustain their war effort against Russia during the Russo-Japanese War. The Rothschilds, along with other bankers, such as Jakob Schiff, head of the firm Kuhn, Loeb, and Co. (more recently part of Lehman Brothers, now Nomura Securities International) supplied financing to the Japanese, in part due to the high rate of interest they would receive but in retaliation for anti-Jewish pograms in Russia. This sort of business led to rumors that the Rothschilds were instigating conflict for financial gain. This is of course ludicrous, as the Rothschilds were primarily invested in sovereign government bonds and loans, and it is well known that war would significantly reduce the value of this sort of portfolio. I will point out that other rich families that were more directly involved in the weapon industry, such as Dupont, that were not Jewish, did not receive nearly the same amount of fearmongering as did the Rothschilds.

It was also around this time that the Rothschilds of Britain became involved in several distressed and startup companies. A distressed company is one currently in financial peril, usually because it has imminent financial obligations that it cannot meet (for instance, a loan payment is coming due, or their supplier is asking for payment). In a distressed scenario, companies are more willing to sell their assets for cheap, or to pay even higher rates of interest. This is due to the much higher risk for an investor or lender (because if the company fails anyways, the investor or lender is vulnerable to losing their money). In this case, the Rothschilds were able to coinvest in a series of extremely profitable projects like the Rio Tinto copper mines (which Rothschild supplied funding to, allowing it to grow and become one of the world's largest mining companies) as well as DeBeers Diamonds, now one of the largest diamond companies. This is essentially the modern equivalent of investing in Apple before its massive turnaround and Google in the 1990s. Put into this context, it is hardly surprising that the Rothschilds were able to do well here, as well as command a reputation among the wealthy that they could be trusted to manage your money.

Regarding the World War I rumormongering, elos_ gives an excellent explanation of the reasons for the war here. It should be quite conclusive that it had nothing to do with any financier and instead had to deal with Great Power tensions.

This would be a bit easier to tackle if you provided one or two references or quotes for this theory - so, books, maybe online articles or any kind of text or video where "the Rothschilds did the world wars" is stated. This way, whoever wants to deal with the question doesn't just have these pretty broad claims to go on.