This is a very good question.

It will help us here if we take a close look at the sentence involved. I've changed the pronoun *he* to the proper noun *Bob*:

 - Bob wants [to go to the market].

Now if you just take a quick look at the sentence, it will seem as if there's just one big verb phrase with the Subject *Bob*. However, if you think about it more slowly and carefully, you will notice that the verb *want* is taking an infinitival "clause". This infinitival clause means something quite different from the verb *want*, and we understand this clause as having its own Subject. 

This is difficult to see here, because we understand the Subject of the verb *want* and the subject of the  verb *go* as being the same person. Bob is doing the *wanting* and Bob will hopefully be doing the *going* too. But if we change the subject of the infinitival clause, this will become clearer. Compare these two examples:

 - Bob wants [Brenda to go to the market].
 - Bob wants [ ____ to go to the market ].

In the first example we see that the Subject of *to go* is Brenda. In the second example, where there is no expressed Subject, we understand the subject of *to go* to be the same as the Subject of *want*. We could model this sentence like this:

 - Bob<sub>(i)</sub> wants [ ____<sub>(i)</sub> to go to the market ]

Or maybe like this:

 - Bob wants [ <s>Bob</s> to go to the market].

So in this last example, we might want to model the clause as having a gap in the subject position, which refers to the same thing as the subject of the main clause. Alternatively, we could just say that the subject is understood, but not expressed.
Whichever way we model the sentence, we still have not only a verb phrase, but also an understood Subject, and if this subject is not understood, the sentence cannot make any sense.

This is one of the main reasons why we consider an infinitival clause to be a clause instead of just a VP. however, there are other more complicated theoretical reasons too.

Hope this helps!