Why does Eugene from the Walking Dead speak in this manner?

TL;DR:

In-Universe: Because Eugene is socially awkward, didn't have many friends for most of his life, and may have Asperger's Syndrome

Out-of-Universe: Because Josh McDermitt is imitating his little brother, Zach.

It is pretty clear that he's trying to impress people, and he's incredibly socially awkward.

Josh McDermitt confirms this:

I think Eugene didn't really have many friends, in the physical sense. I'd say most of his friends were online, in the virtual world. Eugene probably spent a lot of time reading. Being somewhat of a recluse. And eating Cheetos and drinking Code Red, hahaha!
- Reddit AMA

And:

“He’s definitely a weird guy,” says McDermitt, who based Porter on his brother, in terms of demeanor and cadence. “I’m not trying to bring comedy to that role, but I think there’s something about him that’s kind of comedic because he’s different.”
- Backstage

Executive producer Gale Anne Hurd suggested that Eugene may have Asperger's:

Access: And how Eugene developed his unique style of speaking! You guys must have a lot of fun with Josh [McDermitt, who plays Dr. Eugene Porter]. I've spoken with him before and he's a very funny guy [to interview], but [on screen] he delivers his lines so well.

Gale: Exactly, and it's surprising... Eugene is a very difficult role to play, and there are times when we're on set and we're in complete awe of him, of Josh and the performance that he gives, because you're really buying it that he is Eugene and then he comes over and changes in an instant and is back to being his normal self as opposed to sort of his Asperger's [Eugene] self. The character, I don't know, there doesn't seem to be a lot of psychologists left alive in the zombie apocalypse, so I don't know if he'll ever be actually described as Asperger's, but he seems to have those traits.
- Access Hollywood

Asked to comment on Hurd's statement, McDermitt said "I don't want to address that", perhaps because he has said elsewhere that he based his performance of Eugene on his brother, and doesn't want to suggest that his brother has Asperger's.

We don't know why the writers write his lines the way they do, but we know why he speaks with the inflection and cadence he does:

Q: Have you ever known anyone like Eugene before? Is there anyone that he reminds you of?

A: I definitely know people who are like Eugene; if they don’t know something, you’ll never know that they don’t know it. In fact, I’m hesitant to say this because it will make him upset, but I’m pretty much doing an impression of my little brother when I do Eugene. He’s dry, he doesn’t show any emotion, and you’re thinking, “Man, this guy’s an asshole.” Well, he’s a sweet guy, my brother. But he also will BS his way through anything. He convinces people he’s Russian all the time, and he went out and learned a lot of Russian to be able to speak to people so that he could back up this claim. And look, my brother and I are cut from the same cloth. So it was very easy to identify with Eugene.
- Q&A – Josh McDermitt (Eugene Porter)

And:

Someone asked how I came up with the voice for Eugene: Oh! That... I actually based Eugene's voice on my brother Zach... and he gets mad every time I tell people that! But he speaks with such a flat, even affect, that he was like the perfect model for what Eugene sounds like. Eugene's emotional spectrum is so tiny - I couldn't not steal my brother's persona, to make Eugene...
chuckles
- Reddit AMA

And:

My little brother is just like me in terms of he likes to screw with people too. He was working at Costco as a forklift driver. Now, he’s a travel writer but the whole time at Costco, he had all his coworkers convinced that he was Russian. Like he had this big, thick beard and a Russian cossack hat that he would wear and he would talk about the time when he was in the Red Army. He would just BS his way through work everyday.

Then I get on The Walking Dead and I start telling people I’m basically my little brother and this and that then he calls me up like, ‘Stop telling people that! You’re ruining my thing.’ And I’m like you’re lying to your coworkers. But yeah, he really got mad. The thing is he’s just dry ’cause he doesn’t speak with any affect at all, so I based Eugene on him in that regard but he’s definitely not as socially awkward as Eugene is ...I get so many fans who come up [to me] and go, ‘You sound just like I pictured him in the comics’ and that, to me, is one of the highest compliments ’cause I do not want to be the guy that shows up and screws up the show or screws up someone’s image of these characters in the comics.
- Vibe

And:

[My brother's] regular voice, he has such little affect when he speaks. He would be incredibly excited about something and I would just say, “Zack, are you excited?” He said, “Yeah,” like that’s him extremely excited. I kind of took that element of him and brought it into Eugene, of just kind of this guy who speaks so flatly.
- CableTV.com

And:

Eugene was just really a mixture of several years worth of people that I’ve observed over time. And with the voice, I’m doing an impression of my brother in terms of how little affect he uses when he speaks. I know my brother gets mad when I say that, but I don’t really care. (laughs)
- Journey of a Frontman

And:

I come from a family of people who don't really communicate. We all love each other, but we're just bumps on a log. We sit there at family parties and it's awkward because nobody really speaks. That's all in me. It was easy to pick that up. I've made jokes in the past that I'm just doing an impression of my little brother, because he's extremely dry and extremely sarcastic and extremely smart. So it has just been this fun thing to make jokes that I'm doing him. But it hasn't been too hard, because it has been in me already. It's not like it has been a stretch, and the more I do interviews and talk to people, I'm becoming more extroverted to show people I'm not a weirdo like Eugene would be. Trust me: my neutral face and how my teeth look, I look like I'm angry. So if I say something dry and sarcastic, people are like, "Man, this guy's a jerk." So I purposely try and smile as much as I can or try and have fun, so I can be more approachable and people can come up to me and say "Hello" instead of being like, "Oh, he's angry. I don't want to talk to him."
- Under the Radar

And:

A lot of that stuff I was able to draw from the observation of people that I do on a daily basis and the way he speaks, I sort of took that from my little brother Zach, who speaks in a flat monotone way. Actually, a lot of people in my family talk like that but he takes it to the extreme and he’d be really excited about something and we’d say ‘hey Zach are you excited?’ and he’d just say ‘yeah’ with very little affect. I kind of took that for Eugene’s tone of voice and used that as a jumping off point. All the other little ticks and mannerisms and things like that are just things that gleaned off of people as I watched them in my everyday life.
- No White Noise

Josh's brother Zach, on whose voice Eugene's is based, is a travel writer for The Peasant's Guide. Despite a pretty thorough search, I have yet to find a video of him speaking - here's the closest thing I turned up.

Q: Can we get a sample of your brothers voice?

A: Um... I don't know... looks through phone I might have something on instagram... I'll be pissed if I don't...
- Reddit AMA