How do people come up with unique algorithms?

I'd first recommend that you not put the word 'algorithm' on a pedestal. Every time we write code, we write a new algorithm.

I suspect what you really mean is to ask how we create an algorithm so effective and useful that it gets re-used across the industry and taught in schools? In that case, it tends to happen organically - a need arises due to something new in our world/industry, many people devise their own solutions, but eventually someone devises a novel solution that makes everyone say, "Yeah, that is better.", and it gets shared around.

Often people come up with new algorithms by taking inspiration from existing ones. For example, someone might see a sorting algorithm and think of a way to improve it, or they might see a search algorithm and think of a way to make it more efficient. There's no one way to design an algorithm, so it's often a matter of trial and error until something works well. One more thing: Don't forget that a lot of the time, people don't come up with new algorithms, they just implement existing ones. So while someone might not be coming up with a new algorithm, they might be coming up with a new way to implement an existing algorithm that is more efficient or more effective in some way.

Learn mathematics or a math-adjacent field (e.g. physics) rigorously. The famous algorithms you'll hear about are ones that optimize a certain calculation to require less computationally heavy or fewer overall "steps" to get to an answer. A new algorithm is just a new way to solve a problem or class of problems within a given set of constraints in a systematic way.

An algorithm is just a set process to calculate something. Once there is something new to calculate - a new algorithm will usually be born.

Usually new things that need to be calculated require new calculation steps. These steps chained together into a process is an algorithm.

After achieving a first working version, optimizations are usually put into place. for example chaining previously written algorithms.

Hope this helps.