Think about what dividing means with a concrete example.

Let's say you have 4 apples and want to divide those into 2 piles. So that's 4 divided by 2. You get 2 piles of 2 apples each right?

Great. Now take that same pile of 4 apples and divide it into zero piles. You can't do it, can you? No matter what you do, you're left with some number of piles of apples. Short of making the apples cease to exist, you can't divide them into zero piles, hence you can't divide by zero.

It gets more complicated to use the apple example when you get into negative numbers and such, but this is a straight-forward example for ELI5.

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So if you take 10 items and put them in 2 groups, you have 5 items per group. (10÷2=5) If you take 10 items and put them in 1 group, you have 10 items. (10÷1=10) If you take 0 items and put them in 100 groups, you have 0 items per group. (0÷100=0)

But there is no way to put 10 items in 0 groups, you still have 10 items, so it wouldn't be zero, but they're not grouped at all so it's not a number (like 10), so you get 10÷0 is invalid or null.

Let's look at numbers very close to 0

1/.1=10

1/.01=100

1/.0000001=10,000,000

Etc. The closer you get to dividing by 0, the closer your answer tends towards infinity (and negative infinity if you come from the negative direction.)

So why would you arbitrarily define 1/0 to be 0? It doesn't give you any extra predictive power. It doesn't really lead to much, and you still have to leave it out in any situation where you're not defining it. It's just not a very useful definition.

Simply put, Divide by zero is actually infinity. Although, mathmatically it cannot be solved (leading to calculators giving Div-by-zero error)

You're about to get a crash-course in limits.

Mentally, This can be thought of as taking any number and dividing it by smaller and smaller numbers. The smaller the divide-by number is, the more MASSIVE the answer. If you divide by the smallest number possible (Ten, divided by 0.00(...)01 with nearly infinite zeros, but still 1 at the end, you'll get a nearly Infinite number). basically infinite

10/5 = 2
10/2 = 5
10/1 = 10
10/0.1 = 100
10/0.001= 10000
10/0.00000001 = 100000000
so, continue this:
10 / 0.0000000000000000000(infinity zeros)0001 = MASSIVE 10000(infinity zero)000 number = basically infinity.