Why are home networks prefixed with 192.168?
The short answer
An Internet spec called RFC 1918 reserved a few blocks of addresses for "private" networks, which is what you should use when you don't have enough public, routable IP addresses to go around. 192.168/16 was one of those blocks.
The long answer (and then some)
Back in the good old days, everything on the Internet got its own public, routable, "real" IP address that almost never needed to change. It was awesome because anytime you wanted to run some cool new app that implemented some new protocol, or set up your own server at home that was reachable anywhere, you could just run the software and not worry about network address translation (NAT) port forwarding/mapping or ALGs or DMZs or trigger ports or hairpinning. The engineers writing the software also didn't have to become experts in NAT traversal issues just to make their cool idea work out.
But the Internet Assigned Numbers Authority started worrying that they would run out of addresses too quickly, so they made it harder for ISPs to get addresses. So ISPs made it harder for customers to get addresses. By the time home broadband and home networks really started catching on, it had become common practice to give each household just a single public IP address (and even that's about to go away soon). So if you wanted to get more than one machine in your household onto the Internet, you had to run a NAT gateway that fakes like all the machines on your home network are actually sharing the same single public IP address. But all those other machines on your home network needed their own private IP addresses to use to talk to the NAT gateway, and an earlier Internet Engineering Task Force specification called RFC 1918 had defined a few blocks of IP addresses that could be used for private networks like that:
10/8 (10.0.0.0 through 10.255.255.255)
172.16/12 (172.16.0.0 through 172.31.255.255)
192.168/16 (192.168.0.0 through 192.168.255.255)
NAT actually breaks a very important Internet design principle called End to End. Hopefully we'll be able to get back to uninhibited End-to-End connectivity as we move to IPv6. IPv6 has an address space so big we ought never be able to run out, so we should never need NATs on IPv6. It'll hopefully re-open the door to a lot of innovative protocols that are a bit hard to write when all of the machines that would like to use the protocols are behind arbitrary numbers of differently-implemented NATs that manage to break things in lots of different, often subtle ways.
It is a private block of IPs that aren't allowed to be routed on the public Internet and are reserved for internal use to be NATed to the outside world. The document that defines this is RFC 1918, which is enforced by IANA.
The blocks of private-use IPv4 addresses are:
10.0.0.0 /8 (any address beginning with 10.x.x.x)
192.168.0.0 /16 (any address beginning with 192.168.x.x)
172.16.0.0 /12 (any address beginning with 172.16.x.x through 172.31.x.x)
the short answer is, really, we don't know.
having special range(s) of addresses available for local networks is a useful idea, obviously.
and rfc 1918 may have specified them, but didn't explain why those particular ranges of addresses were assigned, rather than any other. (i.e. 192.168.x.y is not inherently different than 193.169.x.y, except that it is agreed by convention to be used as "private". it could just as easily have been any range of addresses.)
0.x.y.z or 255.x.y.z might have been more obvious choices, but may not have been available. so we're left using a random bunch of meaningless numbers.