Its a Reagan quote but in this context it means that you trust their intentions, effort etc, but its still worth checking.
I think it's differentiating total lack of trust ("all your PRs suck while mine are perfect, so I have to read yours") from universal fallibility ("your PRs are great, but all code has bugs, so we all review each other's code to catch some of them").
Is the phrase "trust but verify" an attempt to avoid damaging self-esteem? There seems to be some emotional/ego-based attachment to the concept of trust.