To send mass emails to a campaign, you need to use the "List Email" lightning feature on the contacts object.
1. Create a new list view on the contacts object. Click the Gear icon in the top right and select "New"
2. Select your desired criteria, and ensure you filter by "Campaign"
3. Once the list view is complete, select the drop down arrow in the top right and select "Send List Email"
4. The recipient field should auto populate with your current list view (if not, select the list you wish to send mass emails to), notice that the related campaign will auto populate on the right hand side.
5. You can either compose a new email from scratch (including attachments and merge fields) or select an email template already set up using the tools in the bottom right corner. Select "Review" to get a preview of your email before you send it out.
6. Hit "Send"