Why did he suddenly agree to let Mickey manage him?

Because deep down, Rocky absolutely respected Mickey and trusted his knowledge as a trainer, something Rocky would need if he would have any chance at all against Creed.

But Rocky was also angry with Mickey for the way he kicked him out of his locker, so before Rocky agreed to hire Mickey train him, Rocky needed to air him out a bit and let Mickey know how much he hurt him. He also let Mickey first beg for the job before he threw him out, returning the hurt and rejection. Rocky got his payback..

If you watch the scene where Rocky runs after and catches up to Mickey after he tossed him out of his house, it's not one of Rocky feeling bad for someone he disliked and just handing him the job out of charity, it was clearly scene of mutual respect and friendship.

A "lover's quarrel" if you will.

Sure, Rocky probably did feel bad after he blasted Mickey, but Rocky always wanted Mickey to train him. Rocky probably realized that Mickey was right when he said Rocky wasted his boxing talent by choosing to hang around with bums and be a leg breaker.

Mickey hurt Rocky, Rocky hurt Mickey back. Once he cleared the air he knew the right thing to do was to give Mickey the job. Lover's quarrel over.

Being the better person (despite not owing Mickey anything) is what makes Rocky, 'Rocky'
Most people (myself included) might have basked in the brief glory of telling the old guy where to go.....but (in real life also) it's a short-lived, minor, petty victory, that'll likely come back to haunt both parties.

It's the entire crux of the movie, that Rocky (despite his failings, predicament and shortfalls) isn't bitter nor resentful (and as it stands, both Mickey *and* Rocky bring out the best in each other) Plus, it doesn't hurt that Mickey thinks Rocky has a legit chance either?

As powerful as Balboa's rant (about his 'lot' in life) is....it's the static, distanced (silent) scene which follows (in which Rocky chases a dejected Mickey down the street, and accepts his offer....thus giving them *both* a chance) that brings a tear to my eye, every time.

Rocky always wanted Mickey to train him. And when he got the Creed fight, he knew he needed Mickey to not only train him, but to organize the rest of his corner for him, like finding a cut man for him. However he never liked how Mickey had treated him the last 10 years, and he was especially hurt by him not only taking away his locker, but the verbal undressing he got when he confronted Mickey infront of the whole gym. So figured he wouldn't ask Mickey, Mickey would have to ask him, and make nice with him.

However, when Mickey showed up at Rocky's apartment, with his hat in hand, the things he was saying and the way he was acting legitimately upset Rocky even more, to the point that he forgot to agree to have Mickey train him. Maybe it was that Mickey wasn't very apologetic, or that he was trying to convince Rocky to ask him to be his manager. So Mickey leaves without a deal being made, and Rocky blows off some steam, before coming back to senses, and coming to an agreement with Mickey.

From what I can remember, Rocky came off as kinda entitled. He was a guy with natural talent but never showed much determination or motivation. Mickey was seeking out guys who were the hungriest that showed they really wanted it, not spending his time trying to get the guys with natural talent to want it. So Rocky saw where Mick was coming from and understood why he wasn't given some special attention. Then he showed how much he wanted it.

The most touching scene in the whole movie IMO. I think between the fight interview, Paulie asking him to help, Gazzo giving him $500, and now Mick coming to him to be his manager, it was all too much attention for Rocky to bear, and his pent up anger toward Mick - even though Mick was showing tough love - just had to come out. Not just venting at his tumultuous relationship with Mick, but exorcising some demons of his own past and self-pity. He got it out of his system and being the caring person he is - realized he needed Mick and Mick needed him. And the music when he comes out of the building makes me cry every time.

We have all been through a fight like that where we just let everything out in anger over someone. then within a minute later after making them feel bad we instantly regret it and want to help them. That is the way I took it