Answer: I'll bet that question is asked by a great many people.

Here I cannot rely on first-hand knowledge, since apart from an unfortunate experience with a saxophone in high school I don't play either woodwind or brass myself. But I can improvise:

It all comes from the nature of wind instruments, which have a fundamental pitch that results from the length of their air column. Other pitches must be produced either as harmonics of that fundamental, or by altering the sound column through the use of holes or valves. Before the development of valves on brass instruments, for example, they could not easily play a full chromatic scale. But even with holes or valves, a wind instrument still has a fundamental key. Modern instruments strive to make all notes equal in quality and do that pretty well, but some keys still remain more natural to the instrument than others. Composers and orchestrators make use of the tonal differences between D and Bb trumpets, Eb and F horns, etc.

Perhaps most of all: a Bb instrument plays most naturally in Bb Major; that's the "easy" key for that instrument and music in its easy key is traditionally written in the natural key, C major. On a Bb trumpet you don't need to press any keys to play the Bb (1st degree of the scale) nor for the higher octaves of F or D (5th and 3rd degrees of the scale). It seems logical for those notes, which form the basic triad of the home key, to be written as naturals in the trumpet part - no sharps or flats - and that's how they appear if the part is written in C.

The practice of writing parts for such instruments in a different key from the one they sound is one of those traditions in music that perhaps is not as necessary today as formerly, but remains the custom partly because it's hard to change a known system when millions of performers have learned it and when the vast library of existing music is written that way. No one is about to rewrite all those parts. The main movement away from this tradition is in conductor's scores; some modern composers have written their scores so that the conductor sees every line in its actual ("concert") pitch - but the parts for the players still are written transposed. If they were written at concert pitch, all those performers would have to relearn the associations between pitches and fingerings. The apposite expression is "ain't gonna happen."