meaning - that ideas which our minds cannot reconcile are mutually destructive, sillier still

It's a little tricky. This is how I understand it. I can't be as eloquent as the author.

Churchill is writing about the sentiment behind Le cœur a ses raisons que la raison ne connaît pas, meaning, in effect, the heart know things that the mind cannot understand (or know). He is discussing reason and faith.

The idea that nothing is true except what we comprehend is silly, and that ideas which our minds cannot reconcile are mutually destructive, sillier still.

The idea that nothing is true except what we comprehend is silly...

It's silly to think that only things we can explain are true (or, per his example, it's silly to believe that miracles can't be real because we can't understand how they happened - something that calls for faith, not reason)...

and that ideas which our minds cannot reconcile are mutually destructive, sillier still.

...and it's even sillier to think that to believe in something we don't comprehend fully destroys all reason (or, it's even sillier to think that reason and faith cannot coexist.)