Well, not quite toddler level. I don't remember how to explain grammar to a toddler. I have enough trouble explaining it to me.
Consider a possible answer to the question.
The man suggests calling a tow truck is the best solution.
The man is doing a suggestion. The man is the subject. His action is to suggest.
The thing he is suggesting is that "calling a tow truck" and "best solution" are the same. That is, he is suggesting that it is true that "calling a tow truck is the best solution."
So the object is another sentence: Calling a tow truck is the best solution.
So let's look at the original question.
What does the man suggest is the best solution?
It is asking for some such sentence as the tow truck sentence. So the subject is "the man." The action is "does suggest." And the object is "What is the best solution?"