We see the target of this in verse 2, “people who think that we live by the standards of this world.”
Some people might think our church needs a business model template and KPI’s (monitoring of key performance indicators).
Others may want to take what is in the bible and reinterpret it within the limits of human logic and philosophy.

We demolish

Paul’s forthright language, “we demolish”, is aimed at these people who would reduce God to the confines of human reason. Human language cannot contain the nature of God and His ways.
The “pretension” here is the building of extensive human theoretical knowledge about God, pretending that can define and set bounds to the nature of God. That contrasts with personal knowledge of Him through living in faith.
Our “weapons that demolish” are the work of the Holy Spirit in both giving us the words to say and making those words effective in the heart of the hearer.

Personal application

Our personal application of this comes in “we take captive every thought”. I read this as applying to my own thoughts, that we take all our own thoughts that would lead us into that prideful human knowledge and logic, and hand them over to Christ, to enable our thoughts to be subject to Him and used for His good purpose.

I think here of the example of C S Lewis, at the peak of academic human knowledge, but who made his thoughts subject to Christ, and they were used to bless people across the world.