LUT is merely an acronym for Look Up Table. The term is also used colloquially to refer to a technique to apply creative looks to footage, as well as the more scientific / mathematical needs of transforming colour spaces. LUTs can be generated off of mathematical functions that are sampled at intervals, or by using tools to extract changes in images.
A 1D LUT at its most basic level is a pure single input to single output transform where a single value in put delivers a single value output. EOTFsTransfer functions such as sRGB's OETF / EOTF and other such decodes / encodes are delivered almost exclusively via 1D LUTs. There are also multi-channel 1D LUTs, where each channel can get back a unique value based on its input, or generic single channel 1D LUTs where all channels receive the same output value given identical input values.
Standard 1D LUTs are commonly used to perform colour transforms along intensity, also known as transfer curves. This can take many forms such as sRGB's OETF / EOTF, REC.709's OETF / EOTF, BT.1886's viewing EOTF, one of the many logarithmic transfer curve encodings, ACES encodings, etc. A 1D LUT is well suited for such application given that many of the transfer curves are nonlinear in naturepiecewise functions and no simple power law will accurately describe them.
There are demons hidden away here though. 3D LUTs, given that they operate on a specifically designed range of values, must be properly mapped in order to both meet the range, but to optimise their application. In this case, since the question pertains to the Filmic LUT set, we need to make sure that our 3D LUT is going to sufficiently match what we saw when we were grading the image. Some 3D LUT formats include the "shaper" compression component inside their actual LUT formats.
Given that the default Blender sRGB OETF / EOTF was never designed for any sort of creative work such as raytracing, it is sub-optimal on a number of levels. As a replacement, the Filmic LUT set consists primarily of two transformations that make work appear more photographic. First, it uses a much wider dynamic range from the scene referred values than the default sRGB EOTF uses. This means your work includes and requires light that more closely matches physical, real-world levels. As a result you'll notice much better radiosity, caustics, indirect bouncing, etc. Second, it uses a 3D LUT to desaturate the values as they ascend to the higher dynamic range, attempting to emulate what happens on colour film or DSLRs, and prevent the broken, wonky saturations that happen as colours get very intense under the default view.