A: simple tasks like walking or speaking, which trains the healthy areas of their brains
Here, the verb in red is SINGULAR.
As a result, which seems to refer to speaking -- the nearest preceding singular noun -- conveying the following meaning:
SPEAKING trains the healthy areas of their brains.
Not the intended meaning.
Eliminate A.

B: simple tasks like walking or speaking that trains the healthy areas of their brains
Here, the verb in red is SINGULAR.
As a result, that seems to refer to speaking -- the nearest preceding singular noun -- conveying the following meaning:
SPEAKING that trains the healthy areas of their brains.
Not the intended meaning.
Eliminate B.



Your understanding is correct.
OA: After suffering a stroke, some patients must relearn seemingly simple tasks like walking or speaking, training the healthy areas of their brains to take over the functions of the areas that have been damaged.
Here, COMMA + training refers to patients -- the agent of the nearest preceding action in blue -- conveying the following meaning:
When patients RELEARN, they are -- as part of this action -- TRAINING the healthy areas of their brains to take over the functions of the areas that have been damaged.
This meaning is logical.
_________________

I second what GMATGuruNY has deftly written above. Since you seemed to want the clause marker to refer back to tasks, I will add that the only way the sentence could do so is by placing a comma after that very word, thereby setting aside the comparative information, but even then, the verb following the clause marker would need to be plural to agree with those tasks:

After suffering a stroke, some patients must relearn seemingly simple tasks, like walking or speaking, which train the healthy areas of their brains to take over the functions of the areas that have been damaged.

Since you cannot negotiate the part of the sentence that is not underlined, you know that either which or that CANNOT refer to tasks in the original sentence.

I hope that helps. Thank you for thinking to ask me about the question.

- Andrew

A) speaking, which trains the healthy areas of their brains to take over the functions of the areas that have been damaged -> which modifies speaking WRONG

B) speaking that trains the healthy areas of their brains to take over the functions of the areas that have been damaged -> that modifies speaking (the verb "trains" takes the third person singular) WRONG

C) speaking, training the healthy areas of their brains to take over the functions of the areas that have been damaged -> CORRECT

D) speaking, training the healthy areas of their brains to take over the functions of the areas that are being damaged -> have been damaged > are being damaged so i pick C

E) speaking, which trains the healthy areas of their brains to take over the functions of the areas that are being damaged -> which modifies speaking WRONG