Why did I become a vampire after I rejected the 'offer'?

What you've done is become a vampire the old fashioned way.

Whenever a vampire attacks you there is a chance that you'll contract Sanguinare Vampiris, you probably picked it up from one of the vampires in Dimhollow Crypt. Like any other disease it'll be listed under 'Active Effects'.

It takes three ingame days to transform into a vampire with the occasional slightly cryptic notification throughout though if you fast travelled all the way from Castle Volkihar to Riften and waited around it's quite possible you missed them.

While you can cure vampirism after it's taken hold through the Rising at Dawn quest the simplest thing to do would be to reload a save before you were notified that you became a vampire and simply cure the disease either with the right potion or just hitting up a shrine.

because I was harmed by vampires during battle?

That's probably it. There are more ways of turning into a vampire. Other than choosing to become one, being attacked by vampires is another way to turn into one.

If you already had Sanguinare Vampiris (check the Active Effects section of the Magic menu), you'll become a vampire three days after you contracted the disease, unless cured (such as with a Potion of Cure Disease). This can happen if you had recently been in combat with vampires. Declining Lord Harkon's gift has no bearing on this.

For more information, see the UESP article on vampirism.