What causes bad smells? The short answer is microbes.
Where do odors come from?
Odors can come from all sorts of things. Good odors, like cinnamon, vanilla, sugar, coffee, and chocolate tend to be from plants. But bad odors, like moldy food, farts, sewage, and of course body odor, tend to come from microbes (bacteria and fungi). Microbes can digest pretty much anything, but during this process they give off all sorts of gases that can be pretty pungent. So wherever you smell bad odors, you can be sure that bacteria are nearby.
Air fresheners can capture odors out of the air, while deodorants and perfumes often just mask odors. Unfortunately, none of these options stops odors at their source so the odors will keep being produced. Therefore, once enough time has passed for the air freshener or perfume to dissipate, the odor is going to be back stronger than ever. That means you have to constantly spray air freshener into the air to keep catching the odors. Or you have to keep rubbing deodorant all over your armpits, or keep spraying perfume and cologne all over you clothes. Not only is this bad for your skin, lungs, an the environment, you still haven't stopped the odors!
The problem with fragrances
The common approach to mask bad odors with other fragrances is a bad habit that is fading away quickly. A large body of research has emerged in the last decade demonstrating that synthetic fragrances can cause very significant health issues. These health issues can be even more pronounced for kids, which is why parents should avoid options that just hide or mask bad odors. Fragrances and similar small molecules have been implicated in everything from epilepsy to the rapid early development of secondary sex characteristics (essentially puberty that can happen even before the age of 3).
Instead of using synthetic fragrances, some "natural" deodorants and air fresheners use natural ingredients, meaning they come from plant or animals. But since these nice smells come from fruit and other plant-based foods, that means they are perfect food for the microbes that cause odors. So it can make the smells even worse over time. Therefore, alternatives to fragrances are necessary, and the easiest way to do that is stop odors at their source.
Because bad odors come from microbes, the easiest way to stop odors is to stop microbes. This can be done in a variety of ways, such as washing smelly clothes, showering and soaping up smelly armpits, scrubbing smelly toys, etc. But these approaches just get rid of the smelly microbes that are already there. Moreover, bacteria can double in number every 10-60 minutes when well fed. This means that even a morning shower won't stop you from having a huge bacteria colony in your armpits at the end of the day.
The best defense is a good offense, and therefore neutralizing microbes before they create smells is the best option to stop bad odors. Antimicrobial coatings have recently gained popularity for this, with silver being one of the most potent antimicrobial agents in existence. That's why silver is often touted as being "anti-odor" as it stops the nasty things that cause odors.