Bucky had probably already suffered brain damage from his fall and so this would’ve made it so much easier for them brainwash him, plus the fact that he was missing an arm probably helped. 

Why?

Because do you really think they had that mechanical arm up and ready to go straight away? No. I highly doubt that. So this gives them some times to destroy all previous memories. But not motor memories - that way he still has his ‘muscle memory’ left. In theory, it’s pretty easy to destroy memories, destroy the synapses - those things that like neurons fire and stuff, they help make make memories, that’s all you really need to know. Side note: If anyone wants the super science-y version, then I’ll go into more detail.  

So basically, destroy the synapses and you’ve got a blank memory. Or you can slowly alter the originally memories and tweak them to have Bucky remember exactly what you want. 

Now, most likely what happened with Bucky was the first one, or a combination of both. We know our boy is a fighter - see the HYDRA scene in the first movie - so more than likely if they tried the second option then they’d realise just how unstable that would make him. Most likely he’d have random burst of rage or confusion, because like I said, it’s not an exact science and for reasons I’ll discuss later, he’d most likely remember the correct memories at some point. 

Okay, back to the first idea. HYDRA basically took should be dead Bucky and saw an opportunity. A highly skilled agent who they’d previously tested on and knew could take what they threw at him. They’d already been trying to develop brainwashing techniques, so why not try it on this guy? So the did, and it worked. Obviously. But how?

Well to create a memory you’ve got a three step process: input — encode — retrieval. Now, for the blank slate, it’s the synapses at the retrieval end that you’re gonna want to destroy, that way he can’t retrieve them and therefore they don’t exist. Theoretically. They’re still encoded somewhere, but you don’t want to mess with that because you run the risk of making it so that he can’t encode any memory ever. And what use is that if you wanna control him? 

So he can’t retrieve any of his old memories. When this is done, you wanna wake him up. He’ll be confused, disorientated and probably angry. Hell, a lot angry. But you’ll be seen as a friend who is here to help him, and guide him through everything. You give him a new arm, and you sympathetically tell him about the memories he’s missing. But you’re also inputing these new memories into his head whilst he’s unconscious, allowing him to encode them, and when you mention them, you’re giving a cue and making him more likely to retrieve these fake memories. 

My theory is that Bucky Barnes doesn’t have amnesia. Bucky Barnes just doesn’t have any memories. 

For HYDRA it’s be easy to separate his brain into three distinct area: the self, the order, and the loyalty. The self is fluid and you can easily create this by altering and giving him new memories. The orders, now this will be what we would’ve seen in the movie. This one is where they input all the necessary information they need to get him to do what they want. His target, fake memories of wrongs they’ve committed, reasons why he needs to do it, and most importantly, link it all back to his loyalty for HYDRA. That loyalty area is a key component because it needs to be so strong that it influences the self and his orders. This’ll be where they spent hours upon hours inputting propaganda into him, creating childhood memories of him wanting to sign up, him signing up, doing his duty proudly. Anything you can possibly imagine. Think extreme religion, and that’s what they’ll’ve done to Bucky. 

Now, like I said all of those memories are encoded in there somewhere, he just can’t retrieve them. And we know that cues help trigger retrieval, and that the more often a memory is recalled, the stronger that is. Bucky has known Steve since childhood, and I bet he’d often tell stories of them. Constantly recalling and strengthening his memories of Steve Rogers. So when he sees Steve, he might not be able to make the connection to these lost memories, but they were strong enough that after everything he’d been through, his brain could make the link that he knew this person.

And then they destroyed the link. 

But Steve Rogers is not someone you forget easily, and the more he talks, and most importantly, his actions towards Bucky helps reactive these synapses. (Which is not scientifically impossible either, think repressed memories, oh and that medicine from that Rise Of The Planet Of The Apes with James DeFranco and Tom Felton). He might not remember Steve, but this conflicting feeling of knowing him and his memories is what makes him unstable, and is what makes him go into hiding to try and find out where this feeling of knowing Steve is from. 

I’ll just let that sink in. Bucky Barnes has had his head messed with so many times that they altered his sense of self, his memories - probably all the way back to childhood - and is probably so indoctrinated with hatred for America that he no longer resists order. But when Captain America turns up, something sparks and all of the is questions. HYDRA is so afraid that Captain America will render their killing machine useless that that they have to erase the particular mission, and reinstall a new one with hatred for Captain America. But it’s still not enough. 

Okay, I’ll just give a quick overview of the second one. Memory is very suggestible and there’s lots and lots of cases where people have remembered things that never ever occurred, and eyewitness testimonies where they’re completely 100% sure that’s what they saw. So already, there’s an inherence flaw within our memory. HYDRA used this to their advantage. Instead of erasing the retrieval like they did with the blank slate method. They alter these emotionally charged memories and fill them with HYDRA. HYDRA everywhere. They’ve changed the way it’s been encoded in his brain, but this probably wouldn’t work and would result in eventually having two conflicting memories and cause so much psychological distress that you’d have to go with the first option anyway. Think of Earth-616 Natasha Romanoff. The KGB altered the memories, she had two different memories and basically freaked out. To put it simply, either way you should read those comics to see what I’m talking about. Just read all the comics.