Because they possess anatomical features typical for cacti, the most obvious and well-known being the presence of areoles, the felt-clad spots from where spines, flowers and new stem segments grow. No other plant group have them. There is more evidence, from DNA studies and the flower structure and other anatomic details (of which I do not dare to speak - it was a long time ago). There is a gradual transition from sepals to bracts to petals, I do not know if it is unique for cacti, but it is at least unusual.

I have often heard people wonder if they are real cacti, but why not? Orchids have a wide range of habitats too, from dry, cold grassland to tropical forests.

They all descended from a common ancestor that was a cactus.

Steve,

As TJ points out, they come from many climates - Opuntia ranges from the Peace River, BC, Canada to the tip of South America, and everywhere in between.

Eukaryote,

No, they weren't, as far as fossils tell us (no fossilized cacti have ever been found). Where do you get your information?

"No, they weren't, as far as fossils tell us (no fossilized cacti have ever been found). Where do you get your information?"

While the cactus family is it a bit too young to make much of an impact on the fossil record, current research indicates that modern cacti descended from the Pereskias.

Cacti live in environments where fossilization rarely occurs, hence the scarcity (absence?) of cactus fossils.

They all descended from a common ancestor that was a cactus.

Why, then, was the common ancestor a cactus?

'Why, then, was the common ancestor a cactus? '

Because they were out of ducks.

"Because they were out of ducks."

Exactly, they certainly didn't evolve from ducks.
All valid taxa are nested evolutionarily. If the orchid cacti didn't evolve from an ancestral cactus, then they really couldn't be be considered Cacti. But their morphological and DNA characteristics clearly put them in with the rest of the Cacti, so the only scientific explanation for that is common descent from an ancestral cactus species.