Why do we use the objective case pronoun "me" in "He can watch me clean the car"?

In 'he watched me clean the car', 'I cleaned the car' is an event that is the object of the verb 'watched'. The form is called 'raising to object'. The sub-subject is in object case and the sub-verb is infinite (tense is implied by the main clause).

"me clean the car" is an infinitive phrase as the object of the main verb. The subject of infinitives is always in the oblique case.

sbj: He
vrb: watched
        obj: me clean the car (infin)

Other examples of infinitive phrases as objects:

My mother made me take the trash out
She wanted us to come
We asked him to go home

As pointed out by @William, the object complement clause me clean the car has me as a subject.
It also has an infinitive verb phrase without the normal to complementizer.
As pointed out by @Ami, this construction is called "raising to object" (or B-Raising).

B-Raising produces a noun phrase that's in the ideal position for it to function as object of the upstairs clause and as subject of the downstairs one as well -- at the end of the first and the beginning of the second. Since there are two parsings available, both can be used, and often are.

• [1 He watched [2 me 1] clean the car 2]

B-Raising also produces a construction that's essentially identical to B-Equi, like

• [1 He told me 1] [2 (for me) to clean the car 2]

where me in fact occurs twice, as the indirect object of told, and also as the subject of clean. I.e, it was me he told, and it was me cleaning the car; B-Equi is why me occurs once instead of twice.

So there are tests for Raising, which are explained in the link above.
And sense verbs like watch have special codicils, like omitting the for-to infinitive complementizer.