How does DayBack show conflicts?
The calendar "shows" conflicts, instead of preventing them. So you'll be able to see that two events overlap for a given resource on a given date / time.
Can I show activities side-by-side instead of overlapping?
Yes. In Admin Settings / Views you'll see a setting for "Show events overlapping"; just set that to "No".
Preventing conflicts in Salesforce
If you want to enforce the prevention of conflicts and warn users when they occur--you'd use a Trigger in Salesforce. This would alert your users to conflicts they're inadvertently creating even when they're outside of DayBack. You'll find examples and trigger code you can copy into your own org in our blog here: Date Range Conflicts in Salesforce