If you just look at the last sentence of this quote, it sounds like an endorsement of bootstrappism -- the idea that victims are victims because they choose to be, and if they would only take control of their life, oppression magically would cease to affect them. This couldn't be more in opposition to Martin Luther King Jr.'s message. The implication of that last sentence out of context would be that if you bow to power it will take you, and therefore it is the oppressed person's job to not bow in order to fix oppression. What MLK was saying is that those with power will not just give you a better life; it takes resistance from the oppressed to make any change happen. "We know through painful experience that freedom is never voluntarily given by the oppressor; it must be demanded by the oppressed."
I agree. While deconstructing oppressive structures is solely the responsibility of the privileged, those with privilege will never give it up if the oppressed just wait around for it to happen. Continuous resistance is the only way. Most people have some form of privilege and are also oppressed in some way; for instance I have white, cis-passing, college-educated, non-disabled privilege as well as some forms of financial privilege. I consider that this gives me responsibility for resisting racism, cissexism, ableism, and class- and education- based discrimination. On the other hand, I face oppression for being queer, gender non-conforming, female-assigned, and fat, and I am marginalized for being a sexual abuse victim, non-monogamous, vegetarian, nudist, feminist, non-capitalist, and non-neurotypical. I also have the ability (though not the responsibility) to educate and inform on the ways in which I am oppressed or marginalized.
I understand that because I am white, I have a power to speak back against things like racism which people of color do not have, and if I do not wish to make things worse, I have to actively resist the system of racism; this is true for every privilege I have. I cannot speak for others but I can speak against oppression, and "activism is my rent for living on the planet" (Alice Walker). I must work against all oppression, because it is all linked, and certainly must not attempt to prioritize that which directly affects me. As MLK said, "Injustice anywhere is a threat to justice everywhere."
Martin Luther King Jr. did not put the onus of change on the oppressed; instead MLK spoke to the oppressed about a practical strategy of resistance. If people of privilege turn to MLK for wisdom, they should consider this: "I have almost reached the regrettable conclusion that the Negro's great stumbling block in his stride toward freedom is not the White Citizen's Counciler or the Ku Klux Klanner, but the white moderate, who is more devoted to 'order' than to justice; who prefers a negative peace which is the absence of tension to a positive peace which is the presence of justice; who constantly says: 'I agree with you in the goal you seek, but I cannot agree with your methods of direct action'; who paternalistically believes he can set the timetable for another man's freedom; who lives by a mythical concept of time and who constantly advises the Negro to wait for a 'more convenient season.' Shallow understanding from people of good will is more frustrating than absolute misunderstanding from people of ill will." - MLK