How are components with contacts under the ICs soldered onto boards?

This is a ball-grid array (BGA) package. Each little blob is actually solder. The component is placed precisely by a pick-and-place machine, and then the whole board is heated up to melt the solder (reflow soldering). The solder will flow to the pads on the PCB by surface tension, and the solder mask (hopefully!) stops it shorting out nearby pins. The advantage of the BGA package is the much greater pin density (and hence pin count) available compared to inline or QFN packages. You will typically find high pin count (FPGAs etc) components only available as BGA as it's impossible to bring out 1000 pins to the edges.

These are difficult to work with for prototyping work as it's nearly impossible to tell if the solder joint has been made. In mass production, this is done by X-ray imaging. You can make a reflow oven at home, but as you say the precise placement is quite tricky by hand, although not impossible.

Reworking is very difficult, you generally have to remove the whole component and reapply the solder blobs to remake it. Typically, vendors offer development boards with the BGA chip presoldered with the pins brought out to header pins, which are much easier to deal with.

You have a few questions in your post, and I think I can help with a couple of them.

Regardless, how is each pin heated up to the proper temp so that the solder will melt and the chip will stick to the board?

You need either a reflow oven, or if you're a hobbyist like myself you can get by using a hot air rework station. I have a rework station. It's basically a hair dryer that can get temperatures up to where solder melts. You wave the wand over the part, the part/board/solder gets hot enough, and the solder melts and does its magic.

Is there a manual way to solder chips like this?

Yes, there actually is! It takes practice and patience, but you don't need to send out to have this work done. It is possible to do it on your own. I do. This is the best video I can find that demonstrates the process. This is using a QFN (quad flat no-leads) not a BGA (ball grid array - the part you mention) but the idea is the same. The pins on the QFN are underneath the chip.

I've done this myself and it does work. And yes, it does seem a little like "black magic" when you do. But very satisfying!