Quotation marks are used to mention the exact words spoken during a conversation.
So basically it can be understood as the written form of verbal communication.
And to talk about things that have not happened yet, we use the future tense.
Specifically we use the simple future tense to talk about things that will start and end in future.
Looking at the alternatives:
--had cut : past perfect tense of the verb
--am
going to cut : future tense; "going to" indicates the intention /
plan
of the speaker; and with that we use the base form of the verb. Please
note that we use "going to" when talking about a decision that was made
prior to the conversation.
--cutted :it is the non-standard simple past tense of the verb "cut"
--cut : base / root form of the verb
The first sentence gives us the idea that the second person was already wearing old clothes, which in turn indicates that he already had the planned on cutting the grass.
Thus it can be safely assumed that it was his plan.
Thus grammatically correct alternative here is "am going to cut".
Therefore the correct complete sentence is:
"Why have you got your old clothes on?"
"Because I AM GOING TO CUT the grass."