How to remove stage from URLs for AWS Lambda functions + Serverless framework?

This is an API Gateway feature/convention NOT from Serverless Framework so serverless can't do anything about it.

API Gateway requires you with a stage and it is appended at the end of your endpoint.

API Gateway endpoints are meant for developers though so it is not meant to be user-friendly.

If you want it to be user-friendly, you can add a custom domain for it. Different stages can have different custom subdomains.

In the local environment, we may use the flag --noPrependStageInUrl when running the dev server: sls offline start --noPrependStageInUrl when using serverless offline. Online, we may set up a CloudFront or custom domain.

Triggered by @dashnug's answer "API Gateway requires you with a stage and it is appended at the end of your endpoint" and another reply that I read elsewhere, I 'solved' the problem by making the stage specification a bit less telling (about which stage environment was referred to) by using v1 as a stage. That also suggests some sort of API versioning, which is acceptable in my case as well.

So, my serverless.yml section now contains:

provider:
  name: aws
  runtime: python3.6
  memorySize: 512
  region: ${opt:region, 'eu-west-1'}
  profile: ${opt:profile, 'default'}
  stage: ${opt:stage, 'v1'}  # A trick to don't end up with "production" or "staging" as stage.

One thing you can do is use a custom domain that you own (e.g. mycompany.com) and map that to your API Gateway. This way, rather than making a request to https://ab1cd2ef3g.execute-api.eu-west-1.amazonaws.com/dev/, you would make a request to https://api.mycompany.com/.

There's a plugin called serverless-domain-manager that makes it much easier to set up this custom domains. Check out this blog post for a full walkthrough on how to use it.