Windows 10's new Phone app is deceptive, because it's not really a fully-functional app that lets you make voice calls from your device. Instead, the Phone app -- which is basically just a call log with a customizable speed dial menu -- is a companion app that feeds directly into the Skype video app.
You can only call people on your Skype contact list, which is different from the people listed in your address book (the People app). To add someone to your Skype contact list, you'll need to open up the Skype video app and click Add contacts. You cannot add Skype contacts from within the Phone app, and there's no direct link to the Skype video app.
To call someone on your contact list, click the Phonebook icon at the bottom of the screen. This opens a separate Phonebook window, which is searchable and organized alphabetically.
Find the person you want to call, click their name, and click Skype video Call to make a voice-only call. To make a video call, click Skype video Video call. Both voice and video calls open Skype video.
The Phone app has two screens: History and Speed dial. To add someone to your Speed dial list, click Speed dial and click the + button to open up your Skype contact list.
Select the person and the type of call (video or voice-only) you want to add to Speed dial.
To delete someone from Speed dial, right-click on their name and click Delete. You can also rearrange your Speed dial list by clicking and dragging names into your preferred order.
To delete calls from your call history, click the ... button in the lower right corner.
Click Select calls and select the calls you want to delete. Once you've made your selection, click Delete to delete the calls from your records. You can also delete individual calls by right-clicking on the call and clicking Delete.