I guess the question is, do you understand the current magnetic stripe payment card system and why tokenization (which is what Apple Pay uses) is way more secure?
Do you understand that with Apple Pay, your credit card information is NOT even stored on your iPhone, nor is it stored on Apple's servers. Moreover, and this is the real kicker, your credit card information is NOT even transmitted to the merchant (that means the store) and is not stored on the merchant's systems.
So the Target and Home Depot credit card breaches could NOT have occurred, had those transactions been conducted via Apple Pay. Why, you ask? Because with Apple Pay, there would have been no credit card information on Target's or Home Depot's systems for the criminals to grab via malware (memory scrapers).
In fact, EVEN IF criminals were to access or obtain the token (which is NOT your credit card info) that is transmitted to the merchant for Apple Pay transactions (and in turn transmitted to the card network and your issuing bank for authorization), the token itself has NO intrinsic value and is useless to any hackers or criminals.
So there, in a nutshell, you have it. Apple Pay is AWESOME. For both the merchant, the consumer (that's you) and the card industry as a whole. There's nothing to "believe in" - its not religion. It is based on known and quantifiable standards, such as this (the EMV Payment Tokenisation Specification):
https://www.emvco.com/specifications.aspx?id=263
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What are you doubts? What could go wrong that already can't go wrong with the current payment card system?