How can raise a downspout drain from ground level (Picture)
Hopefully that downspout doesn't ultimately end up connected to the municipal [sanitary] sewer system -- my local sanitary sewer operator, and I suppose many others, bar this practice because it leads to rain flows overwhelming the sewer treatment plants, and this leads to discharge of untreated sewage.

Anyway, on to your question. That white rectangle-to-round adapter looks like an ordinary PVC downspout adapter to a 3" pipe. You could raise this by inserting a 3" coupler and short bit of pipe. These are likely available at a home improvement or building supply outlet. Some retailers might put these together with the other plumbing products; others put these drainage-related items in the sprinkler or landscape supply section instead.

Loosen the downspout from its mounting straps so that it can be lifted out of the adapter, place the coupler and piece of pipe and then re-install the adapter, and finally trim the downspout to fit the new height.