The response (as you marked R) たけしさんのかばんはそのかばんです means "Takeshi's bag is that one/bag".
You asked このかばんはだれのかばんですか, meaning "Whose bag is this?". This would be responded to with たけしさんの（かばん）です; "It is takeshi's (bag)".
But the correct answer given points to a specific bag (out of possibly several that are present). So the question needs to say, "Which bag is Takeshi's (bag)?", or "Takeshi's bag is which (one / of the following)?". And for that, the question needs to be as shown in the correct answer; たけしさんのかばんはどのかばんですか。 (It could also use どれですか).
Also, as @FelipeOliveira mentions in the comments, the topic of the response is Takeshi's bag. Thus, the topic of the question should also be Takeshi's bag. In your solution, the topic changes to "this bag", which is why it's not correct.
たけしさんのかばんはそのかばんです ➝ Takeshi's bag is that one/bag.
○ たけしさんのかばんはどのかばんですか ➝ Takeshi's bag is which one/bag? (Same topic)
× このかばんはだれのかばんですか ➝ Whose bag is this? (Different topic)
For your given solution:
このかばんはだれのかばんですか ➝ Whose bag is this? ("This bag is whose bag?")
○ (このかばんは)たけしさんの(かばん)です ➝ (This bag is) Takeshi's (bag); (Same topic, even though it's omitted)
× たけしさんのかばんはそのかばんです ➝ Takeshi's bag is that one/bag. (Different topic)
＊注 ー R is the given "response". correct answer is the solution to the problem of the test; which happens to be a question sentence. Too many overlapping terms! .·´¯'(>▂<)'¯`·.