Building huge infrastructure in inhospitable environments is, straight up, not economically efficient. There is no direct economic reason to do this. The actual economics have to be indirect: that means that it doesn't need to be a good idea... it just needs to seem like a good idea to enough people.
The westward expansion in American history was largely fueled by a well advertised notion of "manifest destiny", and provides a good lens to see how this might work. First, you have to convince some segment of the population that it is in their best interest to spend time and treasure on the venture. That treasure is spent on things like wagons or space ships that will get them to their destination, on supplies and materials to build a life once they get there.
Those people who are providing the supplies are the ones for whom the effort is financially feasible - and once the colonization passes a certain threshold these are the people who are most incentivized to keep it going.
These interests will lobby the government to provide incentives to people, will make or influence advertising pimping the idea of "a new life in the off world colonies". Governments or other large institutions actually end up providing a lot of the financial outlay that private investors never would, specifically because it's all loss. (Reference the building of the port of Los Angelas, the NASA missions to space and the moon, exploration of the new world.) The government incentive is simple: cultural, economic and perhaps military control of new frontiers. Once they've made the outlay, though, the tech and infrastructure can bootstrap others following in their footsteps.
But to do all of this you need to advertise and convince people it's a good idea: whether the idea is manifest destiny, a gold rush, a new philosophy or what have you. Once they are convinced then the secondary economy is the one that drives development by selling things to the risk-takers.
Robots are nice, but humans are the whole package. They can be taught, have their own built in tools (hands), can communicate new problems effectively, are adaptable and able to respond to the unforeseen, and most of all invested: they all want to live, whereas a robot isn't even self aware. All of that and they come in a relatively small package, weight wise. Therefore shipping them is fairly cost effective, even with the life support considerations. Once you have enough on Mars, they'll expand on their own - and economics will start up to support that particular biological imperative/manifest destiny.
The other main thing to note is this: the economics of the situation will shift radically over time. Initially the economics will be very, very bad. This is where advertising comes in, to convince people they want to 'throw away' resources on the effort. Then, when the parent economies are reaping economic benefit from gearing themselves to support the effort, they will be the engine that continues the drive. However, over time more and more resources will be embedded on Mars, which will create and satisfy demands on their own. Economic margin will be sought and the economy on Mars itself will start to ramp up (with, of course, the occasional shock and/or setback). This will mark ramping from a reliant-entirely-on-imports to a largely import economy. But at some point it will start to have to pay for those imports itself, and will need to export. At this juncture we have a colony with internal economics, and some external trade.
Whether the colony becomes economically successful will depend on whether this external trade is particularly profitable for them - whether the profit settles on Mars or is owned by external sources, or whether it is sufficiently profitable at all. Very likely, the first few colonies will never cross a threshold to where the economy is self sustaining. They will die, and leave behind two valuable things: resources embedded in built-infrastructure and knowledge. Both of these will decay over time, but newer colonies can capitalize on them because they will be cheap by comparison to building it all themselves or making costly mistakes.
Eventually, with enough tries and enough sunk capital a colony will cross that threshold. If nothing sufficiently profitable was found it will cross that threshold simply because the cost-to-stay has been driven so low that staying is a 'cheap bet to make'. If something profitable was found, a boom period would occur (think gold rush) driven by that industry, but not necessarily only about that industry (think Levi's and pickaxes).