Why did large denomiations exist in the early 20th century? In the early 1900s, you could buy a newspaper for 5 cents, a pound of apples for 10 cents and the average salary was about \$40 a month.

Nowadays, an issue of the NYT costs \$2.50 , a pound of apples can cost \$1.15 per pound and the median monthly income is about \$2,200.

Today, it makes sense to have a \$100 bill, we can spend it all on a meal at a restaurant and you would need a few of them to pay rent or make a car payment. But in 1900, I can't imagine anything that could be bought with a $100 bill. Perhaps rent? Or a worker's wages?

Compared to the average salary in 1900, a \$100 bill would be the comparable to a $5,500 bill. I can't imagine anything costing that much money today that would justify producing such a highly valued denomination.

My question is: why did we produce \$100 bills in the early 1900s, when there was probably no practical use for them? And also, did people carry $20 bills around in their wallets or were these considered big money?
There were plenty of relatively large purchases that were regular enough to buy with large amounts of cash. If you wanted to buy a Model T in 1909 you'd need \$825-\$1000. A typical house could easily cost a few thousand dollars in an age when mortgage finance was much less common. So there were plenty of expensive things to need significant cash for, at least occasionally. 

But why cash? The most important part of the answer is bounced checks. If you need to pay someone a lot of money by check you have problem. If Abby gives Betty a check and Abby walks off with the goods and the check bounces, Betty will have a problem. If Abby pays Betty but doesn't give Abby the goods until the check clears, Abby may have a problem if Betty never shows up with the goods after the check clears. You can do escrow, but that's expensive and complicated. Cash allows instantaneous settlement.  

Other options existing in this era have other problems. Cashier's and certified checks, as well as money orders may serve in some cases, but don't easily allow negotiation because they denomination is fixed at the time of issuance. Wire transfers have existed since the age of the telegraph but like today remain an expensive technology for everyday use. 

Hundred dollar bills are the (update: second) most common denomination today, which may surprise you:

Source: New estimates of U.S. currency abroad, the domestic money supply and the unreported Economy

This is mostly because the dollar is used internationally for business transactions.