It means going to the limit of what is allowed, the line represents some limit/law/threshold that cannot be crossed. In this case the veil of secrecy that separates what is/is not public about nuclear weapons.

Normally you would stay well away from said "line". Occasionally though someone may "walk" right up to the "line" but no further.

You can take it to mean that someone knows something secret but is carefully only talking about what isn't secret. The risk is that they might inadvertently reveal some information of what is beyond the line.