To the uninformed believer, the claim that the Church is the pillar and foundation of truth can be perplexing. There are three things we need to understand when we read this passage. First, what is the Church? Second, what is a pillar? Third, what is a foundation?
The word "Church" is particularly important to define. There are multiple different uses of the word "church". We could say, for example, "I am going to church". In this case, the church is the building. But this is actually a more recent thing. Just as there are many congregations today that don't actually have a specific building (mine, for example, used to meet in an auditorium in a council building), many historic congregations lacked any kind of specific building. In fact, in the early days of Christianity, it would have been dangerous to even try to erect such a structure. With the faithful being aggressively persecuted, it was essential to hold services in secret.
So, evidently, Paul wasn't referring to the buildings. Not that this would make a lot of sense anyway, given the context in which it is written. Rather, Paul was speaking specifically of the Church as an institution. Many false institutions, such as Roman Catholicism, like to read themselves into this. Catholics claim that since Paul says the Church is the pillar and foundation of truth, all true Christians must come under the authority of the Catholic Church.
An obvious rebuttal is that other organisations do the same. For example, I once had an Orthodox believer tell me "Catholics were part of that Church before they wanted papal superiority and were kicked out almost 1,000 years ago." See, both the Catholic Church and the Orthodox churches have the same origins stories. The religions were effectively formed during the great schism in 1053. Thus, the fact that they both use the identical argument to prove their case means they have to add another layer to prove themselves over each other.
Christians need not do this, because we do not use this verse to try to prove the authority of a specific organisation. Whereas false institutions try to read their name in front of the word "Church", it is clear from the context that the Church Paul refers to is the body of believers. I am a Christian, therefore I am the Church. If you are a Christian, you are also the Church, even if you are from a completely different denomination (which, if you have a denomination at all, is guaranteed, since I claim no affiliation with any of them). In fact, there are true Christians in every denomination, including the heretical ones. Some Catholics are Christians. Some Orthodox are Christians. I even know a Christian who was a Christian in the Mormon Church (though of course he has since left). A denomination, really, is just a name. Claiming the name of a denomination no more requires you to accept its doctrines than claiming to be a Christian actually makes you one.
This means that the Church that is the pillar and foundation of truth is actually us as believers. We must therefore figure out what this means. We have a responsibility, therefore we must come to an understanding of what that responsibility is. So, what does it mean?
In short, it means it is our job to preserve the truth we have been entrusted with. A pillar is a supportive structure. It holds something up, usually a roof. The word translated "foundation" in the HCSB is hedraiōma (ἑδραίωμα), which likewise means prop, or support.
Think of it this way: Let's imagine all Christians disappeared from the world tomorrow. What would happen to the truth? The answer is complicated. Obviously, barring the rapture, the Church disappearing from the world is quite impossible. On top of this, the Bible makes it quite clear that God gets done what He wants to get done. His word will not return to Him void (Isaiah 55:11), He gives the responsibility of those who fail Him to those who will succeed (Luke 19:11-27), He can achieve the same purpose through different means (Esther 4:14), even a rock can do His will if humans fail to do so (Luke 19:40). Thus, it is quite clear that the Church being the pillar and foundation of truth is a privilege afforded to us by His grace as much as it is a responsibility. If we disappeared, the truth would still be spread by false teachers who at the very least appeal to God's word, and failing that, nature itself would testify of God's truth.
But if we ignore all that and pretend the Church is somehow special, what would happen if we disappeared? Answer: The same thing that always happens when we're not around. False churches would be free to spread their lies and suppress the truth. Unbelievers would never even hear the Gospel, much less have the chance to accept or reject it. And who would publish the Bible if not the Church? We don't have such a large number of Bibles today because ink dropped from the sky. We have it because generations of Christians have been copying, translating, and distributing it far and wide.
This interpretation is supported by other claims Paul made, both in this epistle and elsewhere. Paul begins 1 Timothy by exhorting him to oppose those who teach unsound doctrine. In 1 Timothy 1:3-6, we read "As I urged you when I went to Macedonia, remain in Ephesus so that you may instruct certain people not to teach different doctrine or to pay attention to myths and endless genealogies. These promote empty speculations rather than God’s plan, which operates by faith. Now the goal of our instruction is love that comes from a pure heart, a good conscience, and a sincere faith. Some have deviated from these and turned aside to fruitless discussion."
This, along with other very important parts of 1 Timothy, is actually rather devastating to those who would claim this verse in support of their denomination. These denominations usually teach different doctrines, pay attention to myths, and use endless "genealogies", of sorts, in an attempt to prove their Church goes back to the Apostles.
Fast forwarding to chapter 3, in which the claim that we are the pillar and foundation of truth appears, we see the context in which it is mentioned. Paul is describing the qualifications of church leaders, which it should be noted the Catholic Church especially contradicts. Whereas Paul declares that these leaders must be husbands of one wife (v2, 12), Catholicism teaches the celibacy of the clergy. This causes two major problems. The first is that one can hardly be the husband of one wife if one is not permitted to have even one wife No matter how you interpret it, the option of monogamous marriage must be on the table. The second is that just a few verses later, Paul condemns the requirement of celibacy as a doctrine of devils (1 Timothy 4:1–3), which is hardly a ringing endorsement.
Further evidence of the stewardship interpretation of 1 Timothy 3:15 comes from Paul's other epistles. 1 Corinthians 3:11 tells us that no other foundation can be laid than that which has been laid, namely Jesus. We need to be careful how we build on Him. Elsewhere, we are told that Christ is the chief cornerstone of the household of God, which is built on the foundation of the prophets and apostles (Ephesians 2:19-22). Furthermore, we are living stones being built on top of that (1 Peter 2:1-5), which is quite solid evidence for the priesthood of all believers. We see, then, that the sense in which the Church is the pillar and foundation of truth is in the sense that we are built on top of something else. Future generations will be built on top of us. We are built on previous generations, which are ultimately built on the prophets and Apostles, who are built on Christ.
The ultimate foundation, then, is Christ. If we deviate from Him, we deviate from our stewardship. We are stewards of the truth (1 Corinthians 4:1), not the source of it. Not even an Apostle had the authority to change the truth (Galatians 1:6-8). Rather, our charge is to expose the works of darkness (Ephesians 5:11), oppose unsound doctrine (Titus 1:7-9), destroy arguments that set themselves against God (2 Corinthians 10:5), stand against man-made traditions (Colossians 2:8) and earnestly contend for the faith that was delivered once, for all, to the saints (Jude 1:3).
And how do we do all this? Through the word of God, of course! (2 Timothy 3:16-17). And this is the major failure of those organisations that would cite this verse to establish their own authority. Such organisations universally oppose what they call "private interpretations" of the Scriptures. Whereas true pillars of the truth, like Paul, actively encouraged us to test all things (1 Thessalonians 5:21), and gladly submitted himself to the test of Scripture (Acts 17:11), false teachers seek for you to think of them beyond what is written (1 Corinthians 4:6). They seek for you to forego your responsibility as the pillar and ground of truth because they know that if you test them against the Scriptures, they will fail. Do not fall prey to such liars. If someone cannot prove their beliefs from the word of God, they are not getting those beliefs from God.