The Fed doesn't want layoffs – at least not directly. I'll give you the accepted economist explanation rather than my own, but I would note there is a serious lack of independent thinking in economic circles. It's like these people are all educated in the same elite universities reading the same textbooks – as an outsider the level of group think in economics is utterly bizarre.
Now that disclaimer is out of the way...
So the Fed has three mandates (in theory), minimize unemployment, keep inflation low and stable, ensure moderate interest rates.
Inflation in an economy is always caused by supply and demand dynamics. If there's a lot of demand for something but not enough supply then prices will rise until an equilibrium is reached. Rapid changes in prices is something the Fed is mandated to tackle.
The Fed as a central bank can't really do anything to change how many goods / services businesses are supplying so all they can really do to control prices (inflation) is influence demand.
I won't get into the details of all the ways they can do this here, but to summerise they basically have a few really crap tools, with the primary one being to increase interest rates – not 100% true, but close enough. This has the impact of making it more attractive for consumers to save, rather than spend (basically).
More saving and less spending means lower demand for goods, and therefore inflation should fall back to target. The opposite is true when the Fed wants inflation to be higher. The Fed wanting more inflation was why interest rates have been so low in recent years. One of the consequences of lower demand is that businesses may need to slow hiring, or cut jobs.
Their mandate on employment and moderate interest rates largely stems from low and stable interest rates. This is because employment and interest rates are mostly a product of growth dynamics and the only way the Fed can encourage long-term growth is by ensuring prices in the economy are stable and accommodative to long-term decision making.
When people say the Fed wants unemployment what they mean is that the Fed wants people to stop spending money so they stop bidding up the price of goods and services in the economny. Historically tight labour markets (as we have now) correlate well with robust economic demand, and therefore higher inflation. This is partly because consumers have jobs and can spend, but also because in a tight labour market businesses have to compete for labour so consumers tend to get pay rises easier too.
I'm skipping over a ton here, so feel free to ask if there's anything you're unclear on. But basically the Fed wants you to lose your job so you'll stop spending money and so businesses don't have to compete as aggressively for labour. In theory, this is good for long-term growth dynamics.