Ask HN: How to figure out why free users don't convert?

I have a SAAS app that has a steady stream of new sign-ups for a free trial, and (in my obviously biased opinion as the creator) many more features than competing apps.

Despite this it very low conversion rates, and many of the people who do end up paying seem to drop off after a while. Clearly there is an issue, but no one uses the feedback options I provide in the app either, so it's hard to know what to do.

How would you go about investigating this and finding ways to improve conversion rates?