National Rail Fares website - why do fares vary from todays date to a week?

Advance ticket fares change the way air fares do, but the standard fares are always the same (there can be peak and off-peak standard fares).

There are a certain number of seats allocated on each long distance train for "advance" cheap fares. In peak hours very few, if any, seats will be allocated for advance fares. Once tickets are sold for all the advance seats on a particular train, then they're gone and you have to pay a higher fare, either "off-peak" if it's the right time of day or a weekend, or else the phenomenally expensive "anytime" fares.

As said, those are advance fares. These are limited offer non flexible fares (you must travel on the date and time train booked), you save money by giving up flexibility and choice. The more expensive fares are called walk-up fares and are always the same price (except for inflation related rises of course)