Based on encountering thousands of touring cyclists on the road, I estimate that only about 25% of people doing a bicycle tour use a trailer. Most use panniers.
How do people afford it? The answers are as varied as the number of people doing it. As far as getting the money, probably the most common answer is the simplest: live frugally and save up for it. As far as getting the time, some people take time off without pay, some quit their job and get a new one when they get back, some save up vacation for years. A decent percentage of touring cyclists are retired, between school and job, between jobs or self-employed. Others have flexible employment that allows more vacation. Very few touring cyclists are rolling in cash. In fact, most are quite poor, because they have traded the consumption-oriented lifestyle for the open road. Likely fewer than 0.1% have a sponsor. Getting a sponsor is very rarely a realistic approach. Some try to cover their expenses by collecting money for a charity. This does not often work well and is not very fair to the charity nor to the donors if you are siphoning off donations to pay for your vacation. Some might set up a GoFundMe account, but you better have a very compelling story if you expect many donations.
How much does it cost? You have a lot of control over that. Most cyclists probably fit between $10/day and $150/day. The $10/day cyclists would never pay to sleep (you can usually find a free place to sleep if you work at it and don't mind some slightly uncomfortable situations) and eat all their meals using grocery store food. The $150/day cyclists stay in motels and eat in restaurants.