I'll try my best:

This text shows the network interface config for a device. Ip is 1.2.3.7 in the subnet 1.2.3.4/30

IPs are 32-bit numbers that represent the addresses, machines in a network use to communicate. Each IP is part of a network. A subnet is a defined subset of a network and represented by a number of 0-32 showing the number of bits at the end, that can be altered.
For example your home network is probably in the subnet of 192.168.0.0/16
This means that your IP starts with 192.168 and the 16 shows the size of the subnet. IPs in that subnet are 192.168.0.0 all the way to 192.168.255.255.
All devices in this subnet are local. You could also make smaller subnets inside this one to structure you local network or (and that is why companies use this) to allow devices in the same network to connect to each other or to disallow devices in different subnets.

To do this, a gateway is brought into the network. Its purpose is to connect multiple networks. At home your router can do this but in a company there probably is a dedicated firewall-machine.

The gateway also needs an IP. So in this "joke" the server is in the net starting with 1.2.3.4 and only the last 2 bits can be changed. So the IPs are 1.2.3.4 to 1.2.3.7.

The first IP (1.2.3.4) is reserved for the network, the last is broadcasting (1.2.3.7)and we also need a gateway (1.2.3.5) so we only have one IP left (1.2.3.6) so this device is really "secure"...

A Firewall could do the same and a lot of other things as well, but some companies are weird and networking is hard™.

Btw.: I think his 1.2.3.7 should be a 1.2.3.6