The last time Clare Balding came to La Trompette, Chiswick's chi-chiest restaurant, she tells me in a hushed voice as we sit down, was for an anniversary dinner with her civic partner, Radio 4 newsreader Alice Arnold.

"I booked it and I wasn't feeling very well," she explains, "and I ordered a kir royal and then said 'I don't think I'm going to make it.' I got up and I literally just got outside and chucked up."

So happy memories, I say, and she beams back across the table while studying the menu. It's a typical Balding anecdote, frank, self-deprecating and unfussily to the point. Her bestselling memoirMy Animals and Other Family is filled with similar tales of comic deflation.

When we meet, the book – which records her childhood living next to a racing stables run by her father, the former royal racing trainer Ian Balding – is at number five in the bestseller lists. A couple of weeks before she had mentioned in an interview that she hoped to be number one at Christmas. Was she still optimistic?

"No," she says, as if it's a crazy idea. "If it's top five I'll be thrilled."

She says that she hadn't intended the original comment to be serious, and she was regretful at first that she had identified her competitors for the top spot – among them, her old schoolfriend, Miranda Hart.

"I said to Caitlin Moran [another of the front runners], I can't believe they did that, and she said, 'Oh, get over yourself.' And she was right."

After the summer she enjoyed, getting over herself must be a full-time job. By popular consent she was the stand-out TV presenter at the Olympics, and she then went on to repeat the feat at the Paralympics. The feelgood factor surrounding the triumphant summer seemed to settle on Balding in a manner that only gold-medal athletes had experienced.

Newspapers raved about her down-to-earth enthusiasm and sporting knowledge. She says that although her mother kept all the cuttings she hasn't yet had a look at them. "The article I did see, and it made me laugh, was the Jan Moir one." The notoriously tough Daily Mail columnist wrote an effusive piece entitled Why Can't Everyone Be Clare Balding?

"I would quite like to have that laminated," she says, "just to make me laugh every day."

She protests that the praise she has received is unlikely to last, yet she has found herself inundated with work offers when, before the Olympics, her diary was all but blank. To say that she made the most of a special opportunity would be an understatement.

In many ways, Balding is not the most obvious recipient for popular approbation. As her book details, she came from a privileged background of boarding school and gymkhanas, and were she a more obvious product of those trappings, she could easily come across as removed or self-centred. Instead she cuts across all barriers of race, class, gender and sexuality as if they weren't important, because – when it comes down to it in sport as in life – they aren't important. Many of us may know that, but with Balding it's as if she embodies the knowledge, and never more so than during the Olympics.

"How lucky to have been there through two such incredible events which I think made us all better people," she says. "All the negativity seemed to stop. It was like it was OK to be bright and sunny. That suited me too because quite often it's unfashionable to be as shamelessly enthusiastic as I am."

Her enthusiasm isn't conspicuously evident when it comes to food. She likes to eat well but it's not a priority. In her book she recalls the Queen coming to eat – or rather not eat – a meal prepared by her mother. I suggest that cooking for the Queen is the sort of thing to induce a breakdown in most mothers. "Actually," she says, "I think my mother was much more nervous when she cooked for Clement Freud."

She says she's not much of a cook herself, roasts being her strongest suit. She believes concentration is the key to happiness, and that's why people enjoy cooking. "But I'm not happy cooking because it doesn't make me enjoy concentrating."

As a result, she's inclined to buy ready-made meals. Notwithstanding the royal visits, she thinks that she was brought up to think of food mostly as "fuel". "I really love good food occasionally, but I need time to enjoy it and I need to be hungry."

Fortunately we have time before her next television meeting, and she's hungry enough to order two courses. Jerusalem artichoke soup with a poached egg in the middle, followed by grilled salmon. La Trompette used to be much more carnivorously hardcore when it opened a decade ago. But now it's much more in the modern European tradition – tasteful rather than adventurous.

Balding says that she and Arnold don't dine out too often because it's rare that both of them are not working. When they do eat out it tends to be Thai food, her favourite or at Charlotte's Bistro, a more informal place on the other side of the high street. Much of their social life revolves around live animals in the park rather than dead ones on the plate. They meet their large circle of dog-walking friends at dog-friendly pubs.

As the title of her memoir implies, dogs and horses have played a central role in her life. She began riding as a toddler and enjoyed success as an amateur jockey in her late teens, becoming champion when she was 19. She spent a lot of the time either starving herself or deliberately throwing up to maintain what was for her an unnaturally light weight. In no small way, she made the effort to impress her father, who had been a jockey, and who once admitted he liked her more when she was thinner.

Although her father could give John McCririck a run in the chauvinist stakes, she adores him, and he is enormously proud of his daughter and her book. She gave him the chance to remove any aspects that he found too unflattering, but he made no such demands. Instead he sent the manuscript back marked "brilliant" with a kiss.

Still, he can be a trying figure and a couple of years ago she got so upset at him that she broke off communication for a while. "I wasn't talking to him," she recalls with undisguised fondness, "and I swear to God he hadn't even noticed. Three months later I asked my mum, 'Has he realised that I'm not talking to him?' 'I don't think so,' she said. It's sort of a constant test. But quite an interesting one."

I wondered what it was like when she first took Alice to her parents' house. But she didn't take her home, she took her racing. "The first time Alice met my mother was at Windsor races, and I wrote my mother quite a dramatic email after that – she said I deserved a Bafta for it. She's quite good at popping my balloon."

In any case they all ended up getting along, and her parents and family now adore Alice. A couple of days after our lunch, the couple were heading off on a long-overdue holiday, which she described as "unashamedly culture-free", to Arizona and Las Vegas. They were staying on a dude ranch and then going to play golf and have a flutter. It was not hard to imagine this most earthy of women amid the faux backdrops of Vegas, because wherever she is, she doesn't know how to be anybody but herself. That's what makes her such a fine broadcaster. It's also what made her such a fun lunch companion.