The scene with the quote between Tolson and his debaters is the scene after Dr. Farmer has the unfortunate and unnerving incident with the man and the man's pig. Since Dr. Farmer's car killed the pig, the man forces payment from Farmer, then forces him to "help" get the pig in the man's truck:

[Man] Whoa, whoa, whoa! Where the hell do you think you're going?
You got to help us get this hog in my truck.
Come on. Grab the tail end of that, boy.
[...]
Town niggers. They think they're too good to get their hands dirty.

The scene immediately cuts from Farmer to Tolson's living room, where the debaters are all assembled. In a juxtaposition of these scenes and events, the man with the pig is symbolically associated with the debate opponent; the one who doesn't exist; the one the debaters are to be better than; the one who is only a vehicle for words that deny truth: "he is merely a dissenting voice to the truth l speak!" Tolson has the students repeat the recitation, yelling loudly, to enforce in their minds the truth that they are to be judged by one standard only and that is the standard of truth, truth well spoken.

The scene following the debaters reinforces this symbolic treatment when, under the operation of "trying to get to know each other," Tolson responds to a question about his own life with the recitation of the life philosophy of one Willie Lynch, a merciless, "vicious slave owner in the West Indies." Lynch was asked by the colony of Virginia to "teach them his methods," methods gruesome, heinous and violently manipulative ("Keep the slave ... psychologically weak .... Keep the body, take the mind"). Tolson explains that, since by implication the Willie Lynchs stole all the students' minds, he and the other professors on campus were dedicated to the project of giving them their minds back, giving them their ability to reason and to speak truth back:  

I...and every other professor on this campus are here to help you...to find, take back, and keep your righteous mind...

The reason Tolson has them repeat this refrain so incessantly is to train them "to find, take back, and keep" their minds. They will do this by knowing the truth (it is the judge who "decides who wins and looses") and speaking the truth (an opponent "is merely a dissenting voice to the truth I speak!"). The claim for this analysis is that the writer and director juxtaposed three scenes to create symbolism linking the debaters' "opponents" (those in debates and those on the streets with pigs) with the vicious colonial slave owner Jack Lynch. The youths of the debate team will prevail over the Jack Lynches and opponent debaters because they will know truth and they will speak truth.

I am going to go a bit further than the previous answer and say that the key is the very last three words (and punctuation):  "the truth I speak!"  What this means is that a debater must believe, ... further, ... must KNOW that the words he speaks are the truth (no matter WHAT the debate is about).  There are no other sides.  There is only a "dissenting voice," which is not a true opponent. 

Now we must go even further and realize the CONTEXT of this speech.  The debate coach, Melvin Tolson, must lead his African American team to victory, ... in the aftermath of the Civil War reconstruction:  the 1930s.  At the time, the African American fear of white people was very, very real.  Lynchings (and other hate crimes) were common. 

The African Americans of Tolson's debate team, then, were actually AFRAID of their opponents for their own safety as well as the muted (and erroneous) fear that whites were better than blacks, even at debating.  This was Tolson's attempt at getting rid of that fear.  God:  the only judge.  Truth:  the only winner.

Professor Tolson seeks to get his debaters to understand that they are not competing against an opponent.  Rather, they are competing against a higher level of argumentation that transcends their opponent.  In this light, Tolson is trying to get his debaters to understand that they must be arguing and competing on a much more elevated level, one that appeals to the intrinsic natures of rhetoric, argumentation, and analysis.  Tolson seeks to impart to his students/ rhetoricians that their opponent is secondary to the larger configuration of debate, and, perhaps, secondary to themselves.  This mentality allows them to not be overwhelmed by debating at Harvard and facing the members of that debate team.