For the same reason "free trade" lowers the price of goods. If you can buy X from all over the world/continent competition to win your business will mean you get a lower price compared to just being restricted to buying from the smaller selection of businesses in your host nation.
So we are talking about having more buying power in a larger market? Like we do as part of the EU?

Likewise, being able to employee people from anywhere means you get more people interested and they compete for the job by asking for lower wages.
The concept that wages are driven by availability of people to do it is pretty common knowledge isn't it? It's why nurses get paid less than footballers or why a driver with an HGV license can demand more than one with a standard one.
Yes, of course this makes sense in your average salaried position but in the context of the original post talking about making the poor suffer, implying low end, minimum wage jobs, it doesn't hold true. This is what I was specifically getting at.
Put it another way, if you restrict immigration and people willing to work in jobs at the low end because their pay expectations are too high, it would just force companies to outsource to foreign manufacturing or foreign call centers etc. more so than they do now.