Why does the app need access to my contacts?

In order to sync your contact accounts, the app needs access to your contacts.  The app will not store your contacts on any server, and all syncing is done directly from your iPhone offering you superior privacy.  See privacy policy:
https://playaapps.zendesk.com/hc/articles/115006325267

The app may prompt you to grant it access to your contacts from your iPhone Settings, and if it does, simply turn on "Contacts" as shown below:

If you don't have the option to turn on "Contacts," you may have restrictions turned on.  To fix it, go to your iPhone Settings > General > Restrictions, and either completely disable restrictions, or turn off restrictions for "Contacts" only: