Free Teir limit and how it works

Yes your understanding is correct, you'll start incurring charges over and beyond your Free tier usage. The app getting shut down because of storage limit should not be related to free tier limitations, you can simply allocate more storage to your instance to avoid such failures. Set budget alarms to keep a track of your free tier usage. https://aws.amazon.com/free/?all-free-tier.sort-by=item.additionalFields.SortRank&all-free-tier.sort-order=asc&awsf.Free%20Tier%20Types=*all&awsf.Free%20Tier%20Categories=categories%23storage

Free tier gives you ability to familiarize yourself with AWS services at no cost and use it up to some extent, however it doesn't limit the usage, you can always use the AWS resources beyond free tier, it's just you'd be charged for that utilization which is above free tier limit.

For example, t2.micro EC2 eligible for free tier and first 12 months, free tier covers 750 EC2 hours per month, if you use 800 t2.micro EC2 instance hours in a given month, you'll receive bill for 50 hours for that month.

Your app is shutting down, because it may not have required resources, which it needs and you may want to increase those resource capacity, such as EBS, higher config EC2 for which you'll pay if it's not covered under free tier or exceeds free tier quota. Free tier would never bring resources down because of excess usage(beyond free tier).

For every beginner with AWS, specially using free tier, I'd strongly suggest to setup AWS budget alert to get notification as your budget quota reaches. Remember AWS budget is not real time and you may still get billed for excess usage before budget alerts you but would definitely avoid big surprises situations.

If you want to setup alert, you can consider setting up AWS budgets and configure alarm on those, which would notify you, if you cross the defined usage threshold. Follow the Well Architected Lab instructions here for setting up budget alert based on your usage.