When Catholics speak of a “visible” Church, we don’t mean that you can see it.
We mean a Church which has a divinely-established earthly authority. A “visible” Church (such as the Catholic Church and the Greek Orthodox Church) has authority on earth. Somebody is in charge (somebody who is a Bishop and has valid Apostolic Succession). When a dispute arises over some matter, there is an authority that we can go to to get a definitive answer - which has been assured by Jesus to be true.
We can see the visible Church operating in this manner in the Book of Acts (chapter 15). The issue of circumcision was causing controversy - Jews entered the New Covenant after having passed through the Old Covenant. Could Gentiles just skip the Old Covenant (ie, circumcision - which is how Jews entered into the Covenant) and go straight to the new?
This is a reasonable question, and persuasive arguments can be made for either answer. In a visible Church, there is an earthly authority who can answer this question - and this answer is binding upon all members of the Church. The visible Church held a Council to make this decision (and the visible Church continues this practice today).
What you CANNOT find in the NT is any evidence of an “invisible” church. Paul, for example, clearly thinks of himself as in-charge of various churches. The Corinthians are not free to believe and practice any differently from the Galatians or the Romans. Simon, for example, was wrong in offering to pay Peter to ordain him (Acts 8:9-24), but he was not wrong in knowing that Peter had that authority.
Here is the account of that Council:
[bibledrb]Acts 15:1-35[/bibledrb]