IP Addresses are the addresses used to determine where a packet on the Internet is going, so that routers can decide how to send it along to its next hop. They're also used to identify where the packet came from, so that the other end knows how to address its reply packets, so that they go back to the server that started the exchange.
The initial design of the Internet didn't pay much attention to source IP addressess; all routers need is the destination, to determine which adjacent router to send the packet to. But as the Internet has matured, it has become more complex, and more fully used by non-technical entities like governments and banks, and the reasons why a packet might appear at a router with a source address that router does not have a connection back to have increased.
These reasons might be valid -- an end site with a smart, 2-port router which knows how to do load-balancing, and sometimes send packets out its port B with the IP address of its port A, which belongs to a different Internet Access Provider -- or they might be invalid; they might be evidence of either a misconfigured endpoint system, or one which is purposefully participating in some kind of attack on a remote system, and does not want the packets it's sending to be identifiable as to their source.
Successfull ingress filtering will block the latter packets while making it possible still to forward along the former.