What does “No pressure” mean when you’re meeting someone new?

In this case

No pressure, used in this way, is meant sarcastically to indicate that something is, in fact, important and may put pressure on you.

You could rephrase it as "no sweat" if you implied sarcasm in "no sweat".

In this case, it is just a mildly humorous sarcastic comment—they're saying that since people make judgments of you before they even process who you are or what you look like, you may feel social pressure the next time you meet someone new.

Another example of no pressure in this sense is:

If you cut the wrong wire when disarming the bomb, it will explode. No pressure.

You could, of course, rephrase it like this, and many other ways. The last bit is sarcasm.

If you cut the wrong wire when disarming the bomb, it will explode. No sweat, though.

Alternatively

You can still use it non-sarcastically—it's dependent on context. For example, if someone's going through a rough, sensitive time and you want to offer them help but not make them feel obligated, you could say:

Let me know whenever you want to talk; no pressure.