Okay my stab at the question:
It all boils down to the term: Collision domain.
If there were no routers then you would need to have all computers connected to the same subnet.
I am assuming all computers are connected through a network of switches instead of hubs due it will limit where a datagram packet is being sent.
For reference:
When a packet is being sent through a hub, that package will be proper gated to all devices connected to the hub and thereby severely affecting the available bandwidth.
That is basically why you have bad WiFi on 802.11 b/g/n networks. Too much noise from your neighbours!
Switches on the other hand limits the traffic to only concern those two devices that are currently speaking to each other, but we still run into problems: Mainly ARP
.
Before two computers can talk to each other, they need to find out how to get in contact with each other.
This happens through ARP broadcasting.
The dialogue goes something like this:
192.168.1.5: Hi! I am 192.168.1.5 and my MAC address is aa-bb-cc-dd-ee-ff what is the MAC address of 192.168.1.10?
(All devices listens and memorises the MAC and IP address and one replies).
192.168.1.10: Hi! I am 192.168.1.10 and my MAC address is bb-cc-dd-ee-ff-00.
(All devices listens and memorises the MAC and IP address).
Now without routers you will get ARP broadcast data packages from all devices connected on the Internet.
That would be a bad thing. Just read up on Distributed Denial of Service attacks.
What happens is you will be flooded with data and unable to reply.
It will basically be a shouting contest between you and the rest of the world. Whoever shout the loudest win - which means you will loose every time.
Also unless you use IPv6, you wont be able to speak with all machines connected today.
There are simply not enough IPv4 addresses available, but that is an entirely different discussion.