how make an membership app Suppose I want to write a membership payment app. In this program, I want to give an address and members should send their membership fee to this address:

How can members prove that they paid the fee ?
What do I have to get from members to prove it?
how app should check and verify the owner of transaction ?

I know there is software for this, but I want to know how this system works.
If you are using WordPress to create, manage the website and something like Woocommerce plugin to manage the subscriptions/memberships, BTCPay will help you accept bitcoin payments: https://docs.btcpayserver.org/WooCommerce/
There will be no address reuse and all your issues will be resolved.

How can members prove that they paid the fee ?

Transaction details, Invoice etc.

What do I have to get from members to prove it?

Reusing address is not a good practice. Membership websites can manage things differently and use new address for each membership payment. Once the transaction is confirmed it is updated in the database for the user automatically.

how app should check and verify the owner of transaction ?

Owner of transaction should have credentials to login and use the membership options. App will check the database and know if user has paid for membership or not.
If you are using something else to create the website and manage subscriptions/memberships, I would suggest creating an issue in BTCPay GitHub repository and share relevant details.