Rubbish. Bettas are more comfortable in a small tank than most fish, as they often inhabit relatively small pools in the wild, are tolerant of poor water quality and able to breathe atmospheric oxygen. However, they don't NEED a small tank.
The modern betta strains have been inbred and are essentially a domesticated fish - just as you couldn't expect a pack of Pomeranians to survive as wild wolves, nor can you expect a domesticated betta to tolerate the sort of tough conditions its ancestors may have survived. The crap you get fed by pet stores about bettas being more comfortable in a tiny jar than a nice big aquarium is utter BS. It's designed to justify their appauling treatment of their livestock to Average Joe who doesn't know any better, and who generally falls for it hook, line and sinker, continuing the cycle of abuse.
Also, the fact that bettas CAN survive these conditions is no excuse to treat them like ornaments or neglect them. A one gallon tank is the absolute minimum for one betta, with 2-2.5 gals being preferred as the minimum. 5 gallons is considered the 'best' size as it is a comfortable size to set off most rooms and make a nice display, without being so big the betta gets lost in it. It is big enough to filter properly without the current disturbing the betta, greatly reducing your workload as regards water changes. It also gives you the option of adding small tankmates like shrimp or snails, or with exceptionally good filtration and maintenance, even some otos or pygmy corydoras.
However, there is no reason a betta cannot be kept in a larger tank. Bettas are not big fish, or particularly active or flashy, so if you have a 10 or 20 gallon tank with the betta as the main feature of the display, it will probably be a boring tank. They tend to get lost in it. It will not be of detriment to the betta's health or happiness provided there is adequate filtration and cover for him, but the appearance of the tank and its interest factor is also a consideration. A 10 or 20 gal, well planted, with a betta and some corys or otos, shrimp, snails, etc - can be a stunning display.
When you start talking about tanks larger than 20 gallons, the betta will not need all of the space available to feel that he has a comfortably sized territory. He will still not tolerate the presence of another betta in it, under any circumstances, but he may not use all of the tank space, preferring to hang around the top at one end. Personally, I think it's a bit silly to take a nice big tank like that and immediately restrict your options regarding stocking by putting a betta in it. They don't like to share their tanks with midwater schooling species. In a 20+ gallon tank, most bottom dwellers would be acceptable tankmates provided they are not disruptive or predatory. However, again you have the problem of the betta getting lost in the tank, so you've got this lovely big tank that looks like it has no fish in it. It's a matter of personal choice, and if you like it that way, by all means go ahead.
The basis of this myth may be the community tank. When 'large' tank means 'well-stocked medium sized community tank', a betta will be absolutely miserable in there. An experienced fishkeeper, who is capable of seeing a large tank without having to cram it full of every fish that it may possibly support, bypasses this little problem.
The other issue with bettas in large tanks is open water. If there is too much open water and not enough cover, the betta may feel frightened and exposed, and may try to treat the whole tank as 'territory' and patrol the whole thing. Larger betta tanks, where there will be plenty of swimming space, are usually better when they are heavily decorated. This also increases the number of tankmates you can add without pissing the betta off.