Think about the "Earthrise" picture from Apollo 8:
http://www.lpi.usra.edu/lunar/missions/apollo/apollo_8/image...Or, "Pale Blue Dot" from Voyager 1: https://upload.wikimedia.org/wikipedia/commons/7/73/Pale_Blu...
Space exploration has given us some of the most profoundly transformative new views of life on Earth, giving us a greater appreciation and a greater understanding of what Earth is, what its value is, and informing our relationship with Earth. Mars colonization will be no different, I suspect.
Not only will it help crystallize our self-conception of who we are and what we're about as a species, it will provide a greater clarity on what it means to create proper "living conditions" here on Earth. On Earth it's easy to take things for granted. Which sometimes means we also allow progress to stall because we perceive things as being just "good enough for now". A lot of that will change with Mars colonization, and for the better I think. On Mars they won't have clean air or clean water without putting in a tremendous amount of work to create them. Which they will do, of course, because the alternative is to live without them, which isn't reasonable. But consider life back here on Earth, where we take our air and water for granted. Where we can approach the issues of a lack of clean water or air in a lackadaisical manner, without it being the highest emergency priority. And that's true not only in the developing world (India or Sub-saharan Africa) but also in the midst of the developed world (the lead tainted water in Flint and elsewhere, Victoria, BC in Canada flushing its toilets directly into the Ocean, etc, etc.) And the way we treat environmental issues as something that our grandchildren might have to finally get around to sorting out, but not something to actually begin tackling right now today.
Imagine what it's like on Mars where the starting condition is a bare rock with zero trees, zero rivers, zero lakes. Will Martians plant forests, will they create lakes and rivers? Of course they will. That'll be one of the first things they begin doing. And those things will be precious treasures to them. Just as with air and water, that simple act will help remind us of the natural treasures we exploit and take for granted on Earth rather than preserve and cherish.
Additionally, the technologies developed on the path of Martian colonization will have immanent applicability to Earth. The BFR alone is a perfect example of that, suitable not just for Mars colonization but also for Earth satellite delivery, and point-to-point travel, among other functions. Colonists will pioneer a lot of new techniques and a lot of new technologies for their own unique needs. But I think we'll find that while those needs are unique, the advances they make will have a lot wider application. One thing that will be necessary on Mars is figuring out how to build stuff using a minimal base of machine tools and industrial infrastructure. That sort of cost/complexity reduction will be just as applicable on Earth. As will figuring out how to use advanced machine tools (CNC, 3D printers, laser/water cutters, etc.) to their maximum utility. Growing high yield crops sustainably. Achieving high efficiency of recycling. And on and on.
