Why is 1 the denominator of a unit rate?

There's no proper reason. It was defined as such. Taking 1 as the denominator we can compare between two things easily like:

Two trains are moving with following parameters

1. First moving 500 km in 7 hr
2. Second moving 450 km in 6 hr

We have to determine who is faster...

Now at unit (means one) hour

1. First train is moving (500/7)=71.42 km per hour
2. Second train is moving (450/6)=75 km per hour

So clearly we can say that the 2nd train is faster.