Acceptance doesn't mean tolerance. Acceptance is giving consent to receive something whereas tolerance is the capacity to endure continued subjugation. These are obviously NOT equal.
Why is acceptance part of healing? Well, you always have two choices - to accept your reality or deny it. Denying it will never allow you the opportunity to heal because you are constantly saying "there is nothing wrong". When you accept that something is wrong you can then begin to seek out solutions and create and action plan.
Example: I had severe knee pain. I had two choices - to accept that something was wrong or deny that something was wrong. Denying there was a problem left me in pain and without a solution. Accepting there was a problem allowed me the freedom to go search out a solution. In this case the solution was major - reconstructive surgery due to a severed ligament.
Imagine if I had continued to deny there was a problem! What more could have been damaged? I would still be suffering with chronic pain! But accepting my condition brought about a change, freedom, and healing.
The same goes for accepting people. Once you choose to accept a person for who they are you can then take the next step and decide if who they are fits within the healthy boundaries you have set for yourself and the presence of others in your life.
Example: you have a toxic friend (you pick your toxin). You have two choices - accept that something is wrong or deny that something is wrong. If you accept there is something wrong, you can then choose whether or not you will continue to allow their brand of toxin in your life. If you deny there is something wrong you have no option but to continue allowing their toxicity to exist in your world.
A clue: you cannot heal properly if there is perpetural toxicity in your life.
Acceptance is the only way to have the freedom to find a solution.
One last thought - acceptance and love are not interchangeable. Many proclaim the phrase "You have to accept me as I am" with the notion being that you must not only accept them as they are but you must also continue to allow them in your life. This is not true. You can accept someone is the way they are and then decide that they are not a good fit for your healthy life. You can love them from afar and be entirely genuine in that love. Remember, you have two choices. Just because someone doesn't like your choice doesn't make that choice invalid.