This isn’t the first time I’ve gotten this question, but I looked and couldn’t find a post where I’d answered it before, so I might as well. 

Syrio took the job precisely because he’s the ex-First Sword. The old Sea Lord died, there was an election, and the new man wanted his own First Sword instead of Syrio, so Syrio needs to find a new job. 

And the problem with being a bravo in Braavos is that there are a lot of bravos looking for work, so competition is fierce and underemployment is rampant (hence why so many bravos run protection rackets or resort to mugging or begging with menaces).

We don’t know whether the Sea Lord died of natural causes or whether Syrio left in disgrace because the Sea Lord died on his watch, but it could well have been the case either way that he found it difficult to find employment in the city, because it’s not the best branding. 

Alternatively, one of the ways you get work as an ambitious bravo looking to make your name is to kill a bravo with an established name, like say Syrio Forel. Now, the First Sword doesn’t run, but Syrio Forel might have grown weary of killing hot-headed young men and decided to move to somewhere that wasn’t something he had to deal with on a regular basis.

As for why King’s Landing? It’s a big city near to Braavos that doesn’t have a culture of bravos, as most of the Free Cities do (remember, water dancers are from Braavos, but not all bravos are water dancers). And while the nobility might not hire him for their sons, King’s Landing has a thriving merchant community (and not a small number of foreign merchants) who might want their sons to learn the blade even if they’re unlikely to become knights.

Also….he could also be an actual dancing master. A lot of swordmasters taught dance as well – footwork, balance, timing, etc. are critical to fencing so a lot of the skills carry over, the training rooms are essentially identical so if custom flags in one area you make up the difference in the other.