Investment, in economics, is defined as businesses buying capital goods that will allow them to produce more consumer goods in the future. When investment accelerates, aggregate supply goes up as firms' capacity increases.
Immigration could help lead to this by reducing the price of labor. As immigrants increase the work force, they might drive down the price of labor. In such a case, businesses will be more likely to invest because it will be easier for them to hire workers to use their new capital goods.
I think you can also argue that firms will anticipate greater demand because of the presence of more people. In response, they will invest more, leading to the same result.
When this happens, aggregate supply goes up. When aggregate supply goes up, so does GDP per capita.