One day in 1942, a few months after his recovery following a near-fatal operation for stomach cancer, Henri Matisse fell in love.

By then in his seventies, he was passing an antique shop in Nice, when he experienced what his countrymen call a "coup de foudre", or thunderbolt - love at first sight.

The object of his affections, though, wasn't a woman. It was a chair.

"I was bowled over. It's splendid" he wrote to his friend, the Surrealist poet Louis Aragon. "I'm obsessed with it."

Here it is, in the first section of Matisse in the Studio, a fascinating new exhibition at the Royal Academy of Arts. Frankly, it's monstrous: a 19th-century imitation of a piece of Venetian Rococo flummery, with a seat and back shaped like cockle shells, arms carved to resemble menacing fanged sea creatures, and rough-hewn legs that look like they have languished for an eternity on the ocean floor.

If ever we required confirmation that love moves in mysterious ways, this is proof. Aragon called it a "gigolo". But, in a series of drawings and paintings displayed alongside, we see how Matisse, wizard-like, turned it into something strangely beautiful.

The culmination of his "obsession" was Rocaille Chair (1946), an important late oil painting. Gone are the flounces and textures of the original. In their place are rapid, lambent brushwork, and a dramatic crop that emphasises the chair's voluptuousness, making its curves bulge against the picture's edges. Its dancing visual rhythm imbues an inanimate object with the charisma of a living person. The transformation is astonishing.

There are many spellbinding moments like this in Matisse in the Studio, which displays around 35 objects once owned by the artist alongside 65 paintings, sculptures, drawings, prints, and cut-outs.

Throughout his life, Matisse had an extraordinarily sensitive, even sensuous relationship with the world of things. The objects he collected - African masks and figurines, Chinese porcelain, a glass vase from Andalusia, pewter jugs and silver coffee pots - weren't necessarily precious, in a monetary sense.

Rather, Matisse valued them for their fetish-like hold over him, which inspired him to work. Whenever he moved studio, he packed his motley collection of objects, which he called his "working library".

In a black-and-white picture reproduced at the start of the exhibition, they are grouped together, like children in a school photograph. Matisse wrote on the reverse of the original, which was taken in 1946, "Objects which have been of use to me nearly all my life."

Five years later, he expanded on their importance: "The object is an actor. A good actor can have a part in 10 different plays; an object can play a role in 10 different pictures."

In a sense, this new show is a sequel to another scintillating exhibition, staged 12 years ago at the RA, which explored the importance of textiles in Matisse's work.

The risk is that it could end up looking like a mess, an incoherent jumble sale of bric-a-brac resembling one of those flea markets that are so common in small French towns.

Thankfully, curator Ann Dumas, who was also responsible for Matisse: His Art and His Textiles, marshals her material with deft skill. The first section sets out her argument with clarity, by juxtaposing individual objects with works of art that they inspired directly. This is where we find the Venetian chair, as well as the vase from Andalusia. At once, we sense what drew Matisse to the latter, an appealing blown-glass artefact with a winning, wonky charm. Thanks to its lop-sidedness, it has an unmistakably anthropomorphic quality, arms resting on plump hips.

Nearby, a pair of silver coffee pots reveals the extent to which Matisse could transform unremarkable objects. In the marvellous Still Life with Seashell on Black Marble (1940), above, he turned one of them into a fiery being, with a priapic handle, seemingly aflame for the the luscious pink shell, reclining like a naked female model, to its right.

Later sections, structured according to theme, move beyond individual case studies. The next gallery, for instance, focuses on Matisse's infatuation with African sculpture, which he started collecting around 1906, when there was barely any market for it.

Today, we might accuse Matisse of colonialism: he disregarded the original context of the pieces that he bought. Rather, he was interested in raiding them for visual ideas: African art profoundly informed his radical remodelling of the female nude.

Throughout, though, the show's most satisfying moments are those which furnish us with direct links. For instance, Matisse owned a Roman marble female torso, which inspired his blue-and-white Forms, one of the plates in his illustrated book Jazz (1947).

Both works, ancient and modern, are sexy; an ingeniously positioned mirror reveals how closely Matisse followed the luxurious curve of the torso's backside. Yet, while the Roman piece is pitted and fragmentary, Matisse's image is a vision of sleek simplicity.

In the final gallery, we find a lacquered wood Chinese calligraphy panel, given to Matisse by his wife, Am??lie, for his 60th birthday, displayed above a selection of his paper cut-outs - much like it was in his studio in Nice. The panel wasn't included in Tate Modern's 2014 exhibition of Matisse's cut-outs, and seeing it here comes as a revelation: I had never reflected, properly, on the now blindingly obvious formal correspondence between his undulating cut-out motifs and the serpentine flow of Chinese characters. Extraordinary.

Matisse in the Studio is no blockbuster: not all the artworks are masterpieces, by any means. That said, I adore Red Interior: Still Life on a Blue Table (1947), in the final room. Someone, somewhere, should devote an entire exhibition to the under-appreciated series of late interiors to which this joyously uninhibited painting belongs. Its delirious black zig-zags were inspired by the "instinctive geometry", as Matisse put it, of Kuba cloths in his own collection - several examples of which hang above it at the RA.

Ultimately, this show isn't important because it creates the illusion of proximity to Matisse the man, by presenting us with the knick-knacks with which he surrounded himself. Its strength lies in its interrogation of the creative process. The thing about artistic inspiration is that it is infinitely subtle and complex - about as easy to grasp as smoke. Matisse in the Studio comes as close as any exhibition could to bottling the essence of the artist's creativity.

Until Nov 12; information: 020 7300 8000; royalacademy.org.uk