The classes serve to close some functions over variables for convenience; the attr_readers serve as destructuring functions for convenience; the += replaces a recursive function for convenience and familiarity. The function with the += doesn't mutate any value and it remains pure from all callers' perspectives.
You're complaining about syntax, but this screencast is about semantics.
You only have to copy if your array is implemented as a simple linear sequence of memory addresses. More advanced implementations don't have to copy everything. E.g., Clojure's vectors (its array equivalent) are effectively O(1) for the common array operations that are O(1) on naive arrays, like indexing and insertion. But Clojure vectors are still purely functional. (The actual time for some of those ops is O(log32(n)), but log32(1,000,000) = 4, so it's effectively O(1).)
The term for this is "persistent data structures", usually implemented via trees, where replacing an object in a vector is implemented by building a new tree, reusing all of the old nodes except the ones that appear in the path from the root to the replaced node. That's why Clojure's Vector is log32; it's a 32 b-tree. (I'm writing this from memory and have little Clojure experience, but I'm pretty sure I have it right.)
Tweets aren't editable so no need to handle that. Deleted tweets can be honored by updating the timeline to be `old_timeline - deleted_tweets + new_tweets`.