Unfortunately, I cannot come up with the score for this piece (though I did find that of his second piano concerto, dedicated to his son Maxim) so you won't get any music examples.
At this time in Shostakovich's career he was writing a great deal of music for film, theater and ballet and was becoming quite well-known as a result. Shortly after completing his opera Lady Macbeth and a set of preludes for piano, he began work on this concerto, his first foray into a large symphonic form since his Symphony No. 1 of several years earlier. Early comments from Shostakovich's ex-composition teacher Maximilian Steinberg referred to the piece's brash hodge-podge of styles which included quotes from Beethoven, Haydn and Mahler as well as stylistic references to music-hall and jazz styles. In the last movement, where the trumpet comes to occupy a role nearly as important as the piano, we hear fanfares that always seem to be on the verge of quoting the William Tell Overture of Rossini--a theme that recurs in his Symphony No. 15.
I think that while the Piano Concerto No. 1 is certainly not the kind of ideological "statement" that might have come from another composer of the time, it has shown itself to be a successful piece over the years, popular with both performers and audiences. It reveals a side of Shostakovich that those familiar with his symphonies and string quartets may not know: the young pianist who, to support his family, played piano in cinemas to accompany silent films (later, he was commissioned to compose scores for silent films). Shostakovich's ballets, film and theater music are very little known, but in this concerto, we get a taste of what they are like.
Again, sorry for the lack of musical examples, but I think you will enjoy this music even in their absence. Here is Khatia Buniatishvili, piano; Rainer Küblböck, trumpet with the Vienna Symphony conducted by Philippe Jordan: