How can a web server handle multiple user's incoming requests at a time on a single port (80)?

A port is just a magic number. It doesn't correspond to a piece of hardware. The server opens a socket that 'listens' at port 80 and 'accepts' new connections from that socket. Each new connection is represented by a new socket whose local port is also port 80, but whose remote ip:port is as per the client who connected. So they don't get mixed up. You therefore don't need multiple IP addresses or even multiple ports at the server end.

From tcpipguide

This identification of connections using both client and server sockets is what provides the flexibility in allowing multiple connections between devices that we take for granted on the Internet. For example, busy application server processes (such as Web servers) must be able to handle connections from more than one client, or the World Wide Web would be pretty much unusable. Since the connection is identified using the client's socket as well as the server's, this is no problem. At the same time that the Web server maintains the connection mentioned just above, it can easily have another connection to say, port 2,199 at IP address 219.31.0.44. This is represented by the connection identifier:

(41.199.222.3:80, 219.31.0.44:2199). 

In fact, we can have multiple connections from the same client to the same server. Each client process will be assigned a different ephemeral port number, so even if they all try to access the same server process (such as the Web server process at 41.199.222.3:80), they will all have a different client socket and represent unique connections. This is what lets you make several simultaneous requests to the same Web site from your computer.

Again, TCP keeps track of each of these connections independently, so each connection is unaware of the others. TCP can handle hundreds or even thousands of simultaneous connections. The only limit is the capacity of the computer running TCP, and the bandwidth of the physical connections to it—the more connections running at once, the more each one has to share limited resources.

TCP Takes care of client identification
As a.m. said, TCP takes care of the client identification, and the server only sees a "socket" per client.
Say a server at 10.10.100.100 listens to port 80 for incoming TCP connections (HTTP is built over TCP). A client's browser (at 10.9.8.7) connects to the server using the client port 27143. The server sees: "the client 10.9.8.7:27143 wants to connect, do you accept?". The server app accepts, and is given a "handle" (a socket) to manage all communication with this client, and the handle will always send packets to 10.9.8.7:27143 with the proper TCP headers.

Packets are never simultaneous
Now, physically, there is generally only one (or two) connections linking the server to internet, so packets can only arrive in sequential order. The question becomes: what is the maximum throughput through the fiber, and how many responses can the server compute and send in return. Other than CPU time spent or memory bottlenecks while responding to requests, the server also has to keep some resources alive (at least 1 active socket per client) until the communication is over, and therefore consume RAM. Throughput is achieved via some optimizations (not mutually-exclusive): non-blocking sockets (to avoid pipelining/socket latencies), multi-threading (to use more CPU cores/threads).

Improving request throughput further: load balancing
And last, the server on the "front-side" of websites generally do not do all the work by themselves (especially the more complicated stuff, like database querying, calculations etc.), and defer tasks or even forward HTTP requests to distributed servers, while they keep on handling trivially (e.g. forwarding) as many requests per second as they can. Distribution of work over several servers is called load-balancing.