Suppose we were to find a way of travelling sufficiently fast that time dilation has a very significant effect. Let’s say we send a probe to Proxima Centauri and back, and while the round trip takes a decade, the probe’s clocks show that only a month has passed.
What does this mean for humanity? How does it affect our future?
It means humanity is ten years older when the probe gets back.
The time dilation affects the probe and it’s contents only.
It means that interstellar travel is possible, and presuming that we only have to go as far as Proxima Centauri to colonize it, not even unreasonable. You would barely need supplies for the trip! However if we can’t get the journey to take less than five real years it will still be tough to run a government interstellarly because few people will want to displace themselves from their family/friends to run out there and check on them - not to mention the mail would take five years to deliver too.
Of course this presumes we have the wherewithal to colonize a foreign system.
I think the OP explicitly acknowledged that last part, but I agree that I don’t understand the question. It means we’ll have to wait a long time for things that in the probe’s reference frame don’t take nearly so long. Which means… I don’t know what it means. Watch the movie Interstellar.
Even for a trip to such a close solar system, you’d need a large amount of fuel for a trip at those speeds, especially given how big a colony ship would be.
Assuming that the means of transporting people without the acceleration and deceleration killing them is also developed to travel at those speeds, it now enables the means for a person to “skip over” a number of years. Kind of do the “Planet of the Apes” (original) or “Interstellar” trick.
Say you have a 2 year old daughter, and because you’re rich enough (and weird enough) you decide you’d prefer to miss watching her grow up, but want to walk her down the aisle at her wedding… being younger than her ! It could be done !
Once we establish regular trade with a far-off planet, we need to deal with the economic implications of distant trade at near-light speeds. Paul Krugman wrote a seminal paper on this topic in 1978:
It would take about eight years for a round trip. Also eight years to send a radio signal and receive a reply. I’m looking into space travel by using antimatter to break the light barrier. I expect to see results by 2061. 
It means depending on where you are going, you can forget talking to anyone you ever knew at home again, and you are on your own, mission control cant help you.
Or you might get home in time to bury her
One thing it means that often gets overlooked in science fiction is that anyone who can build one of these engines now has a weapon vastly more powerful and harder to defend against than modern nuclear weapons. Any drive that easily accelerates a space-ship to a high fraction of c will pretty easily cause a dinosaur-killer sized explosion if you point it at earth. If these are fairly easy to build then proliferation would be a huge problem, and if they’re practical for colonization or resource extraction you would need a lot of failsafes to make sure an enemy or crazy person doesn’t casually create an extinction-level event. You’d see some major changes in Earth governments, and the destructive power of the drive might well motivate colonization with the idea ‘if we get far enough away they can’t kill all of us’.
What, Planet of the Apes isn’t good enough for you? 
The fundamental flaw with all these discussions is they assume the following.
Let’s suppose that best case scenario, there’s another planet almost identical to Earth somewhere out there.
Well, first of all, in order to have a breathable gas mix in the atmosphere and habitable (to humans) temperatures, it would take life to do it. Life would need billions of years to make the place livable. And the issue is that while other forms of carbon based life might be similar to earth life…probability says they almost certainly will be just different enough to cause an anaphylactic immune reaction or fatal infection if you were to breathe some of that air. They would be just different enough to be a lethal threat.
So the colonists would have to live in space suits and sealed habitats and they’d have a heck of a time leaving the planet since the dV requirement to escape atmosphere is really high.
Almost no matter how you slice it, it would be much easier to build a permanent human settlement underground on the Moon, or in orbit somewhere. Fundamentally, what you need for a mostly closed loop life support system is a protected armor of rock shielding, some habitation centrifuges (you do not try to spin the whole asteroid, that’s stupid. You instead build train tracks and have the colony ride on maglev tracks. Or you build balloon habitats and have the colonists sleep in a centrifuge wheel inside the pressured balloon. (basically, the balloon doesn’t spin. Instead, there’s a series of rooms that are made of steel gratings or something, and the whole wheel spins at the hub. There is a counter-rotating wheel with half the mass so it doesn’t cause any net angular momentum change to the asteroid it’s part of.
If I’m doing the math correctly, and if you are successful, you should be able to present your results in the past, not the future.
I believe it’s pretty much the case that, in order to solve the problems of getting people to another planet to live, you first have to get to a point where you don’t require planets at all any longer - so (I think) interstellar travel can never be about colonisation.
I think that that’s assuming transit times in the decades and centuries. This thread isn’t about that but assumes transit times in the range of weeks and months.
Are you not forgetting the option of terraforming? Exterminate the indigenous life and transplant our own? And if it takes a hundred years or more, so what?
I know of one novel that deals with this in detail - Pelligrino and Zebrowski’s “The Killing Star” The Killing Star - Wikipedia
I’d still be 15 minutes late for work.
Realistic terraforming would be extremely difficult and expensive, and would probably take something more like thousands to hundreds of thousands of years, not something like a century. It’s common in science fiction to shoot a missile loaded with algae at a planet then have green fields to land on a few decades later, but that doesn’t appear to be remotely realistic for a vast number of reasons. More advanced ideas using nanotech tend to just gloss over dealing with unwanted chemicals and waste heat, but when you’re changing things on a planetary scale you can’t just gloss over enough heat to melt the surface!
If you’ve got the level of tech to create or completely change a biosphere quickly and without much effort, you’ve almost certainly also got the tech to adapt human bodies to whatever biosphere you find, or just adapt to space and forget about landing on planets entirely.
Indeed not but I don’t think that level of terraforming is required; I don’t know what level is, though.