LaurieB and Laurierace have got it covered and both do a great job explaining. I’ve tried for years to educate some people about claiming races and honestly, horse people who don’t follow racing can be the hardest to teach! My riding friends simply cant fathom plunking down $30k and walking off with the horse post-race because they vetted their horse 6 ways to Sunday and peppered the seller with questions and fretted over every step.
As for the mechanics, trainers who claim must be licensed in the state and in some cases there are other rules, like you must have started a horse at the meet or have stalls at the track or attendant training center. (Most racehorses live at the tracks or at big training centers close to the tracks.) The official entries for all the races usually takes place a few days before and the track publishes an “overnight sheet” which is essentially a list of entries for upcoming races. Most of this is online now. Trainers who deal in claiming horses primarily generally know the local pool of horses and keeps and eye on them. When they see one they are interested in at an appropriate price they might buy him themselves or they approach an owner who might be interested and who has the money. (Why a trainer might be interested in a certain horse could be a whole other set of posts!) The money for the claim has to be in your account at the track. The Horseman’s Bookkeeper handles such matters. When purse money is won it gets deposited there and various disbursements made for payment to jockey and other fees. You must have enough in the account to cover the price of the claim and the sales tax and fees if applicable.
Once a potential claim is identified the trainer might try to get any info on the horse that he can, watch for his jogging in the morning, ask any employees for info but claiming trainers play it very close to the vest and don’t keep loose lipped staff! Trainers with a horse entered that they don’t want to get claimed have been known to spread rumors that he’s been ill or sore. They might pack his standing wraps with ice so his legs look lumpy to observers, they might bring him to the paddock (saddling area) before the race with bandages on etc to deter claims. Once he’s there for saddling the trainer who wants to claim him has a few minutes to get look at him. Usually the claim slip must be in the box in the racing secretary’s office by the time the horses leave the paddock. (This is known as “dropping a slip” on the horse.) Then he waits. If there are multiple slips in the box the secretary’s office could be very crowded.
Barring any issues, the horse changes hands after the race. In NY, a member of the racing secretary’s staff goes out when the horses “come back” for unsaddling and clips a colored tag on the halter of the claimed horse who, if he’s not required to go to the test barn, then goers off with the new trainer’s groom to his new home.
Claiming can be cutthroat and as mentioned claiming trainers need a face like a poker player. A few years ago at Saratoga a very nice high class sometime stakes horse was dropped in “for a tag.” The horse was owned aby a partnership and the whole gang was in town. The claiming price was at least $150k! Well, not only did he not win, but the horse was claimed! I’ve never seen so much gnashing of teeth and clutching of pearls. The new connections were being cursed all over the clubhouse. OTOH, in the early 2000’s there was a popular old claimer who had to have a special diet because he had few teeth after an accident. He was a regular winner and every time he was claimed, the trainer who lost him would pass along the recipe for what he was to be fed! It was an act of kindness and respect to a horse that performed so well for so long for so many.
As for races being competitive, trainers want their horses in the easiest company they can face without losing them. Tracks want evenly matched races with full fields because the make money from the “handle” which is the amount bet. All the bets are pooled and the track skims a % off the top and distributes the rest to the winning bettors. Bettors get better payouts for being right when the fields are big and evenly matched.