In requirements.txt, what does tilde equals (~=) mean?

It means it will select the latest version of the package, greater than or equal to 0.6.10, but still in the 0.6.* version, so it won't download 0.7.0 for example. It ensures you will get security fixes but keep backward-compatibility, if the package maintainer respects the semantic versioning (which states that breaking changes should occur only in major versions).

Or, as said by PEP 440:

For a given release identifier V.N , the compatible release clause is approximately equivalent to the pair of comparison clauses:

>= V.N, == V.*

• Definition in PEP 440
• Complete example here in the documentation

Adding to the existing answers, I think it's very important to also mention that while

~=0.6.10 means >=0.6.10, ==0.6.*

Following is also true

~=0.6 means >=0.6, ==0.*

It's mentioned in the PEP documentation.

That's the 'compatible release' version specifier.

It's equivalent to: mock-django >= 0.6.10, == 0.6.*, and is a tidy way of matching a version which is expected to be compatible. In plain English, it's a bit like saying: "I need a version of mock-django which is at least as new as 0.6.10, but not so new that it isn't compatible with it."

If you're not sure about all this version number stuff, a quick look at the PEP440 version scheme should sort you out!