How a transaction proves its ownership of the its source, which can be one or multiple unspent output?

How a transaction proves its ownership of the its source, which can be one or multiple unspent output?

I read the following from the book << Mastering Bitcoin >>

"As you may notice, the transaction contains an empty scriptSig because we haven’t signed it yet. Without a signature, this transaction is meaningless, we haven’t yet proven that we own the address from which the unspent output is sourced."

I am wondering how a signature can prove that this transaction owns the address from which the unspent output is sourced?

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