Update: I seem to get two kinds of answers and I need to get more clarity on how they relate (or not). What we have so far is, roughly in the lines of:

you have 32 bits, the / indicates how you split it

and (or versus)

You have an address, the / indicates the range around it

Main question

I am looking for a way to explain a notation such as a.b.c.d/e to someone who is not too deep into networking.

Imagine it is a manager type, or even a data scientist who has seen some sites can be accessed by typing an IP address, but he/she does not have too much more familiarity with the field.

What have I tried?

I searched for explanations, and found many but often these things talk about 'subnet masks' which would require an explanation by itself.

Therefore I have tried to come up with a simple anecdote, though I am also open to an explanation that is simple without an anecdote I have found this to work well in the past.

Attempt so far -- Warning, comments pointed out my example has some serious issues

An IP address consists of four numbers separated with a dot between each. You read it from left to right, and each dot adds a level of detail. This is very similar to a real address, think about:

Country.City.Street.Housenumber

When working with IP addresses, we often add something like /24 or /32 at the back. This is...

An ip address consists of four numbers separated with a dot between each. You read it from left to right, and each dot adds a level of detail. This is very similar to a real address, think about:

Country.City.Street.Housenumber

I'm sorry, but this is simply wrong. The dots do not have any significance. They are only there to make a 32-bit binary number (that's all that an IPv4 address is) easier for humans to read. You can't infer any hierarchy from them. If you're explaining IP addresses that way, you're misleading people.

I know you don't want to get into subnet masks or binary math, but there's really no way around it. I'll keep it as simple as I can.

IP addresses have a two-level hierarchy: network and host. Think of area code and number for your phone (at least in the US). Ten digit telephone numbers have a fixed format: 3 digits for the area code, and 7 digits for the phone number. Similarly, for IP addresses, part of that 32-bit binary number is the network address portion, and part is the host address portion.

Unlike phones, IP addresses do not have a fixed format for hierarchy. The number of (binary) digits that make up the network and host portions can vary. Where you divide the bits between the two is determined by the subnet mask (There. I said it).

The CIDR notation tells you how many bits of the address belong to the network portion. For example, /20 means the first 20 bits of the address are the network portion of the address -- the rest (32 - 20 = 12) are the host portion. /24 means 24 bits are the network and 8 are the host (32 - 24 = 8), and so on.

If all you're doing is typing in address into your browser, you don't need to know the binary math. But if you're creating networks and assigning addresses, you need to understand the binary math behind them. Otherwise you will make mistakes that lead to incorrect addresses.

EDIT:

I know that my answer will go over the heads of many so-called "managers", but it's important that YOU understand it so you can explain it better. It you're confused, your explanation will be confusing (and possibly wrong). Based on your scenario, the best simple explanation I can give is this:

IP addresses are divided into two parts - network and host, just like telephone numbers are divided into parts - area/city code and number. The /xx indicates how much of the IP address is the network part and how much is the host part.

I wouldn't go any deeper than that. If you get questions, then you'll have to bring in binary and subnet masks.

You don't need even that detail. Simply treat them as magic values:

Short answer: An IP address has four blocks of digits. There is a standard notation called CIDR where if you put /24 at the end it means every IP address that starts by the same three numbers. So if you put 192.168.120.0/24 that means that access is allowed from every IP address from 192.168.120.0 to 192.168.120.255 so only those within the allowed range is allowed into the system.

Any network/security engineer actually needing to set that will understand that setting right away. If you sidetrack the product showcase into a Networking 101 you will lose time that could be used to present other features, while the technical people that is also in the meeting and that know perfectly about CIDR will be cringing about such waste of time.† And that manager won't need such information, anyway. It's best IMHO to ask them to question you after the meeting / send you a mail if they are interested in the long, detailed answer.

Nothing wrong about being curious and wanting to know more about it, but a product showcase is not the right place, and I suspect in reality such manager probably don't really mind about that detail.

† In fact, this simplified version may lead those network engineers to jump asking if you don't support IPv6, which they are using internally. A more accurate version of that answer would start by “An IPv4 address is formed by four octets...“ which would already get the manager lost in the first phrase.

... how many house numbers the street has.

If you have more than 256 houses in a street, you need to use part of the street number for house numbers. This is only important to know when you want to know if someone else lives on the same street as you because you drop off letters directly instead of giving them to the post office.