In the event logs, I found out, that I have errors in system logs:

• Event Id: 7031, 7034;
• source: Service Control Manager;
• details: The SharePoint 2010 User Code Host service terminated unexpectedly.

I found out, that service account that was running Sandbox code service needs to have access to performance counters on the server. So what you need to do is pretty simple, you need to grant service account access to counters and you can do it by adding service account as a member of local group Performace Monitor Users in Server Manger.