My standard answer to how I understand 4D is to say that I realise I do not understand 3D and go from there. This is glib, but I think it does conceal a trick that I do use. That is to stop trying to see in 4d. There have been perhaps a handful of people who could actually work in a purely intuitive manner in 4d space. For most of us even quite simple questions (such as the intersection of cylinders in 3 different directions) in 3d require a lot of thought.
By giving up on trying to actually see the whole 4d picture it is then possible to bring in many of the tricks (such as the jump from 2d to 3d) and other methods of the abstract. Sometimes you can do a lot simply with linear algebra.
Most of the work I have done in higher dimensional spaces has been considering projection tilings like the Penrose tiling, where the higher space is naturally split into two smaller spaces. So don't forget that as well as looking at 3+1, looking at 2+2 can sometimes be handy, or even a longer list of 2 and 3d projections or views on your object. After all this is how we often end up working on 3d things!