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Computers are deterministic

In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.
_{Deterministic Systems - Wikipedia}

Think of a computer like a cook, who has been handed a recipe (a program) to bake a cake. Computers do not deviate from the instructions (input) in a program, meaning that each cake they bake (output) will be identical.

So how do programs (including games) implement randomness? Kinda like this:

_{Comic #221: Random Number - xkcd}

A mathematical formula is used to "fake" randomness. This is known as a Pseudorandom Number Generator (PRNG), often abbreviated to just Random Number Generator (RNG).

The problem is, being a mathematical formula - if you give it the same input value, you'll get the same output value. So we're left with a problem:

What is the initial starting value?

This is where the "seed" gets its name. It's a seed value, a number used to start the random number generator, so that it may generate future pseudo-random numbers.

In most games, the seed is determined automatically using a few tricks, like deriving a value from the system clock, or the number of frames generated before user input. But this is also why clock manipulation or frame-counting are popular tactics in speedrunning circles - because they can take the "randomness" out of certain actions, to guarantee a more perfect run.

In other games like Minecraft which relies heavily on randomness for it's world generation, they allow you to manually specify a seed to get the "same" world generation.

Math/Comp-Sci crash course:

Just for fun, let's look at a working example of a basic pseudorandom number generator:

Next_Value = (Multiplier * Current_Value + Increment) % Modulus

The Modulo % operator gives us the remainder after division by the Modulus, e.g if 10 / 8 = 1 r2, then 10 % 8 = 2. So let's set some values in the formula, and generate numbers between 1 and 100:

Next_Value = (7 * Current_Value + 5) % 101

We need to seed Current_Value with something, so let's do a seed of 1:

and now, let's try a seed of 42

As you can see, we're getting some "fake" random numbers, based off a seed value! Pretty cool, right?