What does it mean to “grow organically”?

Solution 1:

In this context, for the specifications to grow organically means there is no overarching control or direction. Each developer writes their own tests, without consultation with or approval of a higher authority. Metaphorically, it evokes an image of a tree or plant which grows outward and higher from the bottom up. So the phrase describes a "bottom up" process, as opposed to a "top down" architectural or dictatorial process.

So you see, your final question "Does it mean to grow dynamically with changed requirements or get modified with changed requirements?" isn't really addressed by the phrase. It simply means there is no single authority which dictates the specifications. They will arise (grow organically) from each developer's tests.

Solution 2:

I see a slightly different meaning than ghoppe here, which is more about steady, sustained, fixed-rate growth. The New Oxford American Dictionary somewhat supports this:

organic: characterized by continuous or natural development: companies expand as much by acquisition as by organic growth.