Why should the law care about what I do behind closed doors?

The law shouldn't care (if you are a law abiding citizen) about who you take into the bedroom as long as they are consenting adults. How many guns you own. What you eat for supper. What kind of TV shows you watch. Whether you watch porn or not.

It shouldn't to a point.

Your speech is free. But if it causes malicious harm to someone, you can be sued.

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But what if you're raping little girls behind closed doors? Killing gay men? Watching child porn?

Liberty is not boundless. It does have its limits.

The law doesn't care about what you do behind closed doors as long as it is within the bounds of what the law allows.

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Society sets laws announcing those actions that it deems unacceptable in polite society. If evidence appears that causes a reasonable person to suspect that illegal activity is going on, society should investigate. Of course society might find itself having to jump through hoops by adhering to constitutional law. It cannot just invade your personal space and demand to know what you're up to just because they don't like you.

If evidence arises that you are doing these things behind closed doors, don't you think the government has a moral obligation to investigate?

You can stand on a public sidewalk and take pictures of my house. You can't walk into my house uninvited and start taking pictures.

Does the U.S military really prosecute military staff for cheating on their spouse, or do they close one eye if the cheating does not involve other military personnel since adultery is fairly common? Adultery is illegal in U.S military