Mooz's answer is an excellent summary, but since it relies mainly on Wikia quotes, I thought a canon-based answer would be useful. Here's the relevant excerpt from Mockingjay (Chapter 12, page 170; emphasis mine).

"President Snow used to ... sell me ... my body, that is," Finnick begins in a flat, removed tone. "I wasn't the only one. If a victor is considered desirable, the president gives them as a reward or allows people to buy them for an exorbitant amount of money. If you refuse, he kills someone you love. So you do it."

That explains it, then. Finnick's parade of lovers in the Capitol. They were never real lovers. Just people like our old Head Peacekeeper, Cray, who bought desperate girls to devour and discard because he could. I want to interrupt the taping and beg Finnick's forgiveness for every false thought I've ever had about him. But we have a job to do, and I sense Finnick's role will be far more effective than mine.

"I wasn't the only one, but I was the most popular," he says. "And perhaps the most defenseless, because the people I loved were so defenseless."

Mooz's answer is an excellent summary, but since it relies mainly on Wikia quotes, I thought a canon answer would also be helpful. Here's the relevant excerpt from Mockingjay (Chapter 12, page 170; emphasis mine).

"President Snow used to ... sell me ... my body, that is," Finnick begins in a flat, removed tone. "I wasn't the only one. If a victor is considered desirable, the president gives them as a reward or allows people to buy them for an exorbitant amount of money. If you refuse, he kills someone you love. So you do it."

That explains it, then. Finnick's parade of lovers in the Capitol. They were never real lovers. Just people like our old Head Peacekeeper, Cray, who bought desperate girls to devour and discard because he could. I want to interrupt the taping and beg Finnick's forgiveness for every false thought I've ever had about him. But we have a job to do, and I sense Finnick's role will be far more effective than mine.

"I wasn't the only one, but I was the most popular," he says. "And perhaps the most defenseless, because the people I loved were so defenseless."

Here's why the same didn't happen to Haymitch (same chapter, a couple of pages later):

"Is that what happened to you?" I ask Haymitch.

"No. My mother and younger brother. My girl. They were all dead two weeks after I was crowned victor. Because of that stunt I pulled with the force field," he answers. "Snow had no one to use against me."

"I'm surprised he didn't just kill you," I say.

"Oh, no. I was the example. The person to hold up to the young Finnicks and Johannas and Cashmeres."