I want to know the needs/thinking in photography that leads to it

It starts off with you needing more light in a given situation. Either you getting extremely grainy photos due to low light, or you are in need of a wider lens but can't afford the slower maximum aperture, or you are struggling to eliminate camera shake with slower shutter speeds.

However once you start to learn how to use flash, and you get the flash off camera or at the very least not pointing directly at your subject then you start to realise that it's not just about more light, it's about better light too.

The majority of indoor lighting these days is fluorescent, which is improving but does not yet have the broad spectrum of flash lighting. Far worse are the old style tubes found in office buildings, churches and sports halls - the spectra produced by these lights are very narrow, missing a large part of the red spectrum leading to unnatural green/yellow looking skin-tones. In addition to this the colour changes massively through the 50/60Hz mains cycle giving you very inconsistent results. Flash gives you a very broad spectrum centred around natural sunlight (colour can be tuned up or down with gels for different WB).

See: Do fluorescent lighting and shutter speed create a problem with color cast?

I want someone to tell me why it would be good to have one (or not) and what things I can try achieving when I get one

1. The main thing a dedicated flash unit provides is the ability to bounce the flash of a white ceiling/wall and provide the sort of light you'd get in a studio with a hideously expensive octobox. This was shot with a manual flash hand-held (whilst holding the camera with my other hand):

   

2. Small battery powered flash units produce incredibly short pulses at lower power settings and let you freeze action in ways impossible when working indoors without flash:

   

3. Once you have the flash off camera you can get the flash in unique locations, such as behind the subject, for dramatic effects:

   

4. Get the flash close and you can compete with the sunlight. This was shot outdoors in the afternoon pointing directly into a very strong sun. A pair of flashes close in prevent the subject from being completely silhouetted:

   

The key point here is that in addition to getting a flash, you need some method of triggering the flash so you can use it off camera. Fortunately the prices of portable radio triggers has fallen and reliability has risen significantly. It's just something else you need to factor in cost wise.

I am thinking of cheaper or cheapest flashes when I say flash. Something like YongNuo series

With the exception of one shot, the above images were all shot with a fully manual Vivtar 285. The only thing you get from a more expensive flash is automatic TTL metering. But you don't really need that, part of the learning curve is figuring out how to set the flash power. Plus it's usually less fiddly to just shoot a few test shots and dial in the power, than it is to get a full metering radio setup and mess around with the ratios. I would look at something like the Lumopro LP160, it has everything you need for off camera flash, nothing you don't need and it's cheap.