Q: What's the point in using touch ID?

What do you mean?

Its an alternative security feature where by you can unlock your phone, buy in the online stores, and use Apple pay with your finger print.

The iPhone will scan your prints, and you can use them to authenticate yourself when performing an action that requires authentication.

Its used instead of your passcode, and apple id password.

Have you set your prints up, in Settings->Touch ID & Passcode. 

Use Touch ID on iPhone and iPad - Apple Support

Once you've setup at least one print, you can use that finger to unlock your iPhone.

What exactly happens when you try?

Follow the steps available here: Use Touch ID on iPhone and iPad - Apple Support

That means either the finger you tried is not the one you setup, or the setup was not done correctly, and it can't identify the finger.

Try setting up other fingers and try again.

When you capture your fingerprint, it is best to move your finger slightly so you capture more of it.  That will prevent an alignment problem.

Only if you pair the 5s with an Apple Watch.  Otherwise, no.

https://www.apple.com/apple-pay/