(I'm going to ignore the "Biggus Diggus" for this story =P)
What humanity didn't anticipate was, that those aliens were .... well ....alien. Even the most crazy extraterrestial life forms in SciFi novels and movies and whatever boil down to the same concept: Take one head, two arms and two legs, connect them to a torso, and THEN go wild with extra features like tentacles and acid blood ... oh and keep it vaguely human-sized as well, ok? And they communicate via sound, because that's what language is, right?
The alien that had recently appeared apparently didn't get that memo. Not at all.
The scientist was staring in disbelief at the tennis ball sized and perfectly round sphere hovering in front of him, trying to make sense of the creatures(?) flashing lights, erratic movements and strange odors.
"What am I supposed to do with that?" he asked the sergeant.
"You're the scientist here. You tell me. It's YOUR job to figure out how to talk with ...it. Them. Whatever."
"I don't even know where to start! Maybe.. maybe it's something like ... how bees communicate with each other? You know, they 'dance' and -"
"I'm not going to dance for an oversized, sentient marble" the seargant growled. Meanwhile, the hovering orb flashed bright red and yellow while drifting to the left, over and over again always in the same pattern but more slowly each time, almost like a parent explaining something blatantly obvious to a particularily dumb child VERY slowly and in easy-to-understand words. You gotta start somewhere after all.
The scientist sighed heavily and looked back at the orb. He was always the nerd of the group, the daydreamer, the one with unreasonable expectations about reality, and this situation was something he always dreamed of ... but in HIS version, the aliens were always something like Klingons or Ferengi. Something with a face. Something with a mouth and voice. Nothing ever prepared him for THIS.
He went silent for a solid minute until the orb eventually stopped and turned opaque. The room filled with a faint, sweet scent reminiscent of blueberries and lavender. It was obvious that "it" was desparately trying to make itself understood somehow, but the way humans - and basically all other mammals on earth - learn how to interact with each other, is by mimicking what the older ones do until it starts to make sense. How was he supposed to change color and produce various scents?!
At least he could try to move like the orb ...? The scientist raised his hand, made a fist and slowly moved it to the left. The orb immediatly turned bright pink and the air started to smell like curry for a moment - maybe an exclamation of happiness because the scientist finally started to "speak" with it? Could they even feel happy? So many questions, and no answers yet. He had to think of a way to change color at will... somehow.
The sergeant left the room at this point, leaving the scientist and the orb alone.
It took almost two months until they were able to at least mimic each other. The scientist and his team had built a machine that resembled a cross between a hurdy gurdy and barrel organ with a huge LED display in front, able to spray a couple different scents into the air and make the display flash in the same patterns as the orb, while the orb had somehow learned to shift into a crude, almost grotesque floating head with a face that got closer to the uncanny valley by the day. The main problem the scientists had to face were the fact that the alien seemed completely deaf, so talking via voice wasn't an option. The second problem was, that they had no clue how to quickly replicate various scents AND immediatly get rid of the ones that already lingered. You can only spray artificial strawberry scent into the air so many times until it sticks to everything.
Both parties had to heavily scale down their own 'language' in order to even say hello to each other, but they made progress - albeit slowly.
The biggest surprise however was that the orb simply vanished into thin air one day, only to reappear in the same nonchalant manner within the lab an hour or so later, and this time, it brought something with it - ten smaller, translucent, oval-shaped items, and two hovering plates. The scientists couldn't make sense of it at first, until the two plates drifted forward and the ten ovals formed around them, five on each plate.
"...are those...hands? It's trying to mimic our hands! Quick, get me someone who speaks sign language!"
"You think THEY speak the same sign langauge as anyone on Earth?"
"Oh shut up Frank and get to work!"
It didn't take long after that for both parties to be able to actually communicate with each other ... in a crude, one-word-at-a-time manner with many misunderstandings, but they could talk with the other one. It turned out that the orb was able to display colors that were simply out of the human spectrum (it signed Color - No - See and then patiently waited for the scientists to figure it out) so what the scientists had displayed on the LED screen wasn't even remotely close to what the orb actually tried to show them. It was rather similar to how some colorblind people just can't see any difference between green and red and percieve both as the same color. On the other hand, the orb didn't seem to understand the concept of "sound" at all and had trouble to communicate without the scents.
Then, one day, the scientists asked the question that had been bugging them for months on end: The scientist signed: You - Here - Why
(It had been quite the effort to make the orb understand what "why" even meant in the first place, mind you.)
The orb, this time as a floating face similar to that of the lead scientist, signed back only a single word: Curious.
After the scientist translated the answer back to the audience, the sergeant scoffed. "You seriously mean to tell me that this ... thing ... visited Earth and spent months trying to learn our language just out of sheer curiosity?! Just like that?"
The scientist shrugged. "We did the same, didn't we?" Everyone in the room knew damn well just how many billions of tax dollars the organisation ate up each year, for gimmicks and gizmos that had the sole purpose of trying to find sentient life outside of Earth. And why? Because humanity wanted to know what was out there, that's why. Sheer curiosity.
The sergeant scoffed again. He didn't like it when his subordinates talked back, but he knew that the scientist was right. "Well okay, I guess. Ask it if it's going to take one of us to its people. Its home planet or whatever. We still don't know where the hell that thing even came from."
The scientist faced the alien, pondered for a moment ... their shared vocabulary wasn't exactly big at the moment, and the question was a more complex one than what they had asked the orb before. Eventually he signed: We - You - Home
... and the orb vanished into thin air without giving an answer.
"What the f[%$&] was that about?" growled the sergeant. The scientists had no answer .... yet. It was roughly four hours later that the orb returned, and this time, it brought its "people". Thousands of them. Apparently, the orb had understood the signs as an invitation to stay.
We - You - Home.