Implications of being accepted without funding to a computer science PhD in the United States? I was accepted into Ohio State University CS PhD without funding.

I have heard that people who can show their capability may get funding after getting into the program. Is this understanding correct? 

Is being accepted without funding a dangerous sign? Does this means that the faculty doesn't care about you?
My answer applies exclusively to CS in the United States, or other scenarios in which the standard PhD offer comes with guaranteed funding.
Implication #1: How the department feels about you
First, I will quote from an answer by JeffE (who is a member of the admissions committee at a top CS department in the US) to another question (also about CS PhD offers in the US):

A typical PhD offer from a strong department includes guaranteed funding in some form.

That may come in the form of guaranteed RA/TA work, or something else, but whatever it is will be promised at acceptance. Therefore, the main implication of a PhD offer without funding is that (as you have intuited), the department does not consider you a top candidate for their program. As JeffE remarks in the same answer:

Do not accept a PhD admission offer without funding. If they really want you, they'll pay for you.

You asked: &quot;I have heard that people who can show their capability may get funding after getting into the program. Is this understanding correct?&quot;
It's not impossible to get funding after beginning the program (e.g., if you really hit it off with a potential PhD advisor who has grant money to spare).
But this depends very much on luck and circumstance, not just on merit; so unless you like living dangerously, it's not an advisable strategy.
Implication #2: How it will affect your future prospects
Having said that, if you somehow manage to support yourself while doing a PhD, it probably won't matter to anyone that you were self-funded. Per Suresh's answer to another question:

There's nothing on your CV that needs to indicate exactly how you were supported during your Ph.D.