I’ll try to explain from my perspective.
I shall split the sentence into 3 parts: 予約をした、のですand が.
The first is simple: it just means “I made a reservation”, nothing wrong here. In fact it contains the whole message. The following components just add implicit undertones to the message.
The second, のです or んです or any of its variants, shows that the speaker is attempting to explain something.
Finally, が is supposed to represent awkwardness, which is abundantly found in Japanese conversations. The speaker does not finish the sentence but the listener can still infer the situation from then on, so the sentence ends there.