###EDIT###
iOS 10 and above:
Use Callkit, take a look at call directory extension
According to documentation,
Identifying Incoming Callers
When a phone receives an incoming call, the system first consults the user’s contacts to find a matching phone number. If no match is found, the system then consults your app’s Call Directory extension to find a matching entry to identify the phone number. This is useful for applications that maintain a contact list for a user that’s separate from the system contacts, such as a social network, or for identifying incoming calls that may be initiated from within the app, such as for customer service support or a delivery notification.
For example, consider a user who is friends with Jane in a social networking app, but doesn’t have her phone number in her contacts. The social networking app has a Call Directory Extension, which downloads and add the phone numbers of all of the user’s friends. Because of this, when the user gets an incoming call from Jane, the system displays something like “(App Name) Caller ID: Jane Appleseed” rather than “Unknown Caller”.
To provide identifying information about incoming callers, you use the addIdentificationEntry(withNextSequentialPhoneNumber:label:) method in the implementation of beginRequest(with:).
class CustomCallDirectoryProvider: CXCallDirectoryProvider {
override func beginRequest(with context: CXCallDirectoryExtensionContext) {
let labelsKeyedByPhoneNumber: [CXCallDirectoryPhoneNumber: String] = [ … ]
for (phoneNumber, label) in labelsKeyedByPhoneNumber.sorted(by: <) {
context.addIdentificationEntry(withNextSequentialPhoneNumber: phoneNumber, label: label)
}
context.completeRequest()
}
}
Because this method is called only when the system launches the app extension and not for each individual call, you must specify call identification information all at once; you cannot, for example, make a request to a web service to find information about an incoming call.
iOS 9 and earlier:
As Kakshil mentioned, Caller ID is not possible on non jailbroken devices.
And I will give you some findings on how true caller works,
Its not reading the call history, they actually created Action
extension, wherever you try to share a contact, it will display
truecaller app extension, which coded for showing the contact
details fetched from their server
You might also get confused with push notification received for few
calls, saying
"Some X calls you". This is where truecaller used a trick. If you
noticed clearly, that push notification will be received only when
you get a call from an android user with truecaller installed. Let
me explain you in details,
X(android user with truecaller installed), calling Y(ios user with truecaller installed), the android version notify the server that X making call to Y. And server will send push notification to Y's iPhone.