What should I do if advisor asks me to move to another university or lab for strange reasons?

There are many plausible reasons why the professor may not want to reveal what exactly you said was inappropriate; most likely the professor doesn't want to inadvertently reveal the identity of the person who accused you of inappropriate behavior in order to avoid retaliation or further confrontations.

As a grad student, you are considered an adult and a professional. What matters here is not whether you intended to cause harm, but whether your behavior caused harm. As we grow up the world tends to become less forgiving of our mistakes, and the professor has every right to remove someone from their group to maintain a healthy working environment. Although it is also possible that whatever offensive thing you said was simply the last straw that broke the camel's back, so to speak. Either way, you should reflect on your behavior and tendencies and learn from the mistake. If there are mitigating factors (e.g., being a foreign student, mental health problems, etc.), ask the professor nicely to reconsider... but don't expect anything. Ultimately a successful grad school career depends to a large degree your relationship with your advisor. If the relationship is so broken that your professor is trying to kick you out without any prior warnings (I assume), then there is no point in forcing the matter, no matter how famous the professor is.

Also some assumptions in your question is unwarranted... Why do you think the professor has planted "spies" among the students? More likely the professor learned of the matter because someone reported you. I think you over-estimate how much the professors care about the everyday activity of their students.

The revelations arising from your colleague's discoveries raise several causes for concern, which may lead your supervisor to believe that your application was misleading, made in bad faith, or even fraudulent. The issues as I see them (writing from a UK perspective) are as follows.

Not one, but two references from "friends"

It is possible that a reference/recommendation from a "friend" may be deemed inappropriate, especially a family friend. Having two references/recommendations from friends is even more problematic, even if it were permissible (because it would make me wonder why you are struggling to find anyone else to vouch for you professionally). If permissible, you should declare that the referee is a "friend" (and the reference/recommendation should also declare this). If the referee is a close friend, failure to declare the fact might be deemed fraudulent.

Might the reference/recommendation have exaggerated the extent to which the referee knew the subject?

"we reached out to my parents' friend and also got a strong recommendation"

"reached out" implies (to my British-English ears) that your "parents' friend" was not very familiar with your work, but wrote a strong reference/recommendation as a personal favour. To be honest, he/she should not have written the reference/recommendation in such circumstances, or he/she should have made it clear that he/she had never worked with you professionally. A reference/recommendation should make clear the capacity in which the subject is known to the referee, and not mislead by omission. Although it is the referee's responsibility to get this right, a lapse will, rightly or wrongly, still undermine the subject's credibility.

Irresponsible attitude that may perpetuate nepotism (or perceptions thereof, and thus incite self-fulfilling prophecies)

"My friend and my parents' friend both gave me a strong recommendation to my advisor. And of course, I excelled in the interview. So the connection is very important."

...And of course, people who go into academia with this attitude that "the connection is very important" will only perpetuate nepotism in the future (especially if/when the time comes for them to make hiring/admissions decisions). This comment is irresponsible not only because it impugns your supervisor's capacity to select candidates on merit, but, more generally, because it might discourage people without friends well established in the same academic discipline from applying. In case anybody reading this is thinking of applying for a PhD studentship, I would like to reiterate that you do NOT need to already have friends or family connections in the same academic discipline.

Having said all this, your supervisor is acting irresponsibly in asking you to leave in the manner you describe. If your conduct is deemed so inappropriate as to make your position untenable, he/she should initiate formal disciplinary action. It is possible that your supervisor does engage in nepotism (the fact that he/she had dinner with your parents is slightly incriminating… then again, that happened after you were admitted), and, now that the rumour is abroad, is hoping to get rid of you quietly to avoid getting into trouble himself/herself.

They say that "loose tongues sink ships".

That the situation on the whole is ticklish is not your fault. But talking in a sensitive situation is like smoking in a powder keg factory and throwing the match into a corner.

You may - inadvertently, I am sure - have created a situation where the whole lab of your prof, which is already in a precarious situation, is in even more serious threat to "blow up" and taking parts of the whole department with it.

What's more worrying in your attempts to fix it is that you "don't think it's [your] fault" and say that "[you] just told other people the truth". In sensitive situations, there is rarely the truth, but only approximations thereof. By talking or gossiping, you create the truth, and probably one that neither the prof (and no one else, possibly not even you) wanted.

Your best bet is to go to the prof, ask for forgiveness and explain that you were too inexperienced to understand that your indiscreet talk was totally inappropriate, without ifs and buts. Do not find excuses, do not try to give reasons. Ask to be given one second chance, promising not to gossip in the future.

Do not try to rationalise, do not try to state that you still do not see your mistake. If you still believe it was not a mistake to spread around what you call "the truth", better leave the group right away and start afresh.