How to grow in a start up as a developer?
For sure, the scale is low. Product is easy to understand. How to grow so that I can proudly say that I solved this business problem or at least sound like I solved something good?

"How to grow so that I can proudly say that I solved this business problem" - take a business problem, and solve it?

Pretty much what ToSensei said

It doesn't really matter if you're in a startup or corporate. Arguably it's easier to grow in/with a startup than it is in a corporate environment but the essentials are the same. You become a Yes-man/woman

When there's a problem, be the first in line saying "I can look at it" or if you perceive a problem in your current production then on the next meeting or daily just say it: "We often have the problem X in production, I think/know that the best way to solve it might be doing Y, we should allocate time for it and I can put it together"

If you get a negative response it doesn't matter. Just be pro-active. When something breaks look at it and investigate it and work tightly with other members, especially in other teams.

This is true for any and all jobs, you can't just wait that an opportunity will be given to you, you have to relentlessly push yourself into the forefront of everything.