Yes. Mailspring does need to be running in order for your scheduled emails to send at the time you requested. If you're using MacOS or Windows 10, you can close your laptop / computer and the messages will still send when your computer periodically wakes during sleep to perform background tasks (on macOS, this happens roughly once an hour.)
On other systems, you may need to keep your computer open for emails to send at the scheduled time.
If your computer is offline and unable to send your scheduled email, it will automatically send the next time Mailspring is run.
Why is this?
When we launched Mailspring we conducted a survey and found that most of our users would prefer that the app never send email account credentials to the cloud for security and peace-of-mind. Mailspring syncs your email locally on your computer and does not transmit your passwords to our servers at any time. Unfortunately, this means that if your computer is offline, we can't deliver emails on your behalf from the cloud. We may change this in the future if there's enough interest!