Kate Herron, the director for Loki, has spoken rather at length as to why they went with the aesthetic they did... mostly in out of universe terms:
You mentioned the British influence, and that's really strong in some of the TVA design. I just want to get an idea of what you drew on for the inspirations of bureaucracy and the actual visual aspect of it.
Yeah, when I pitched to the studio, I had this big pitch document. And a lot of it was brutalist architecture, because I grew up in southeast London, and there's a lot of architecture like that there. A Clockwork Orange was filmed near where I grew up, and Children of Men. I wanted to kind of bring that idea into it, because you've got these godly timekeepers overseeing everything. And that felt really appropriate in that regard. But then, the TVA are very heroic, so this kind of Midwest Mad Men-style felt very appropriate as well. And I think building on that, I drew from so many different sci-fi films. To name a few: Brazil, obviously; Metropolis. Our time doors were inspired by Dune. The font on the computers was inspired by Alien.
I think I was drawing from lots of different places, and even my own life - I worked as a temp in a lot of offices and as a medical secretary for the NHS. I remember some of the technology I be using would be really archaic and old, and I was like, "Oh, man, this needs to be updated." But I love the idea that this all-powerful organization that basically controls our destiny, maybe they don't have the most up-to-date technology. Maybe it doesn't look super futuristic, because they aren't necessarily in the future and they aren't necessarily in the past. They're completely outside of that, because it's TVA. It was really fun. I think, bringing that retro-futuristic vibe in that sense to the technology - that it did feel a bit old, but it also had a spin on it that felt a bit different.
Something that Kasra [Farahani, the production designer] and I spoke about was keeping the TVA feeling like this really living, breathing place. We really wanted to build as much of it practically as possible, so I could do long takes with my DP Autumn [Durald]. You see in it the first episode: you have Owen and Tom leave the elevator, and they walk down that long hallway into the time theater. But we built that deliberately, so we could do that. Because I think the more they can interact with it, the more real it starts to feel to everyone.
ScreenRant, Kate Herron Interview: Loki Director
There are also a few other relevant interviews including this other Screen Rant one, this from Entertainment Weekly and this from The Verge. I won't include the quotes here though as the above one is the most complete I feel.
In universe obviously it's not really been explained yet. Firstly, they may look human but they aren't; it's worth noting that a lot of the species we see are human looking or humanoid: Asgardians, for example, could you tell an Asgardian apart from a human side by side? This has been discussed before though, see here and for something a bit less relevant this question on spacefaring humans.
I also want to note they had to have some look. To us, the viewer, it looks very human Earth 60s. If we went to another planet they might say it looks like what they had at some point in their history. There's no indication that the TVA, or that department of it, is designed to look specifically like 1960s Western Earth, that's just how we see it looking from an out of universe perspective.
We may get more information about this as the series develops and the TVA is fleshed out more but for now the out of universe answer is really all we have.
Unmarked spoilers for episode 3 follow, proceed at your own risk
In episode 3 it is revealed that the “employees” at the TVA are actually variants from Earth, they’re just not aware of it. As such it makes sense that the place is reminiscent of Earth to an extent because the people are from there.
Sylvie: I had to pull a memory from hundreds of years prior, before she even fought for them.
Loki: What? What'd you just say? Before she joined the TVA?
Sylvie: Yeah. She was just a regular person on Earth.
Loki: A regular person?
Sylvie: Loved margaritas.
Loki: I was told that everyone who works for the TVA was created by the Time-Keepers.
Sylvie: That's ridiculous. They're all Variants, just like us.
Loki: They don't know that.
Loki, Season 1 Episode 3, “Lamentis”