It is a view through either a
phase contrast or maybe a differential interference contrast microscope.
The bright areas are "phase positive" where the interference is positive (making it very bright) rather than negative (darker than background).
This is a cell culture so it is not easy to tell if the "string-looking-like parts" are axons (outgoing signal path from the neuron) or dendrites (incoming signal path). They are usually called
neurites in cultured cells. Cultured cells are usually grown on a flat surface, which makes it easy to visualize all their parts at one time.
It is a time-lapse movie which is why things are moving fast.
I am guessing they are some kind of
vesicles (membrane enclosed structures).
They are moving fast because they are being "transported" probably along microtubules (pulled by little protein motors that latch on to the vesicle and move along the microtubule).
Can't tell what their contents are, but that might have something to do with their brightness (due to how they interact with the light). Cells in culture are not always "normal" or an obvious member of a well defined cell type.
The
cell bodies (third paragraph) would usually be at the center of an array of neurites. Flat grey areas at the distal end of neurites are
growth cones which put out a lot of small fine processes that "sense" their local environment and direct which way the neurite will be further extended. Growth cones and their fine processes contain a lot of actin. Actin can also move things around, but not so much in the neurites.