If your not a BD member you can still register and log in, but then you have to search for a show the horse was at, find the horse then click on it's name. It then take's you to it's page.
To check how many points a horse has you click on competitions on the menu on the left of the homepage and then on results. You can then search a horse's competition record and a rider's competition record, among others.
To check on the horse's record just click on the link, put in the horses name then click on search. The horse's name will come up with the number of results. If you then click on the horses name a complete list of it's results will come up and the total number of horse points will be shown on the top left (just above the words 'show details')
Unfortunately you only seem to be able to do a horse/rider search if you're a BD member, not with web login status. Having to go round the houses by finding the venue then clicking on the horse/rider link is a bit bizarre - why can't BD let you search like full BD members do (and as per the BE website)? It's not enough of a faff to make me pay to join BD (in case that's their ploy...).