Here is an example:
Now, it is the theme we all know and love, but why do the other bits of the music (the parts that are not the main theme) sound so very much like a movie score to me? Is it just because movie music is decidedly modern in its orchestration and that this is also an example of that? The extra bit of this sounds so very much like a movie score to me and I cannot quite explain why.
The twist here - the composer of the theme singled out in the question, Sergei Prokofiev, actually did compose the scores to multiple movies, possibly the most famous of which is Alexander Nevsky. While that movie was released back in 1938, the film was widely acclaimed, including receiving the Stalin Prize in 1941, so it is very possible that it still influences the movie score-makers of today. At points, Prokofiev even had to comply to the other movie custom of composing music to fit the film footage, which when done too obviously is called "Mickey Mousing".
Perhaps composers who similarly favour orchestras such as John Williams and Hans Zimmer are contributing to this. Williams even admits being influenced by 20th-century classical music composer Edward Elgar when he puts down in the liner notes for the first Star Wars movie about its "Throne Room" theme, "I used a theme I am very fond of over the presentation of the medals. It has a kind of Land of Hope and Glory feeling to it, almost like coronation music".
This style of music didn't arise in movies and get mimicked in symphonies, it's the other way around - this symphonic piece wasn't written to sound like a movie score, it's that movie scores are often written to sound like symphonic music.
Instrumentation is surely part of it. Movie scores are often performed by a full symphony to give a rich palette of sounds and textures to work with, and tend be mainly instrumental as lyrics could conflict with dialogue. Symphonic music also lends itself well to original compositions that blend into the scenes of the movie, as opposed to other styles like pop music which will tend to draw more attention to the artist or music itself, and which cannot be as easily modified to fit the timing or emotional beats of the scene. Other styles of music or instrumentation may also evoke and contribute to a particular setting, like electronic instruments in futuristic settings or traditional instruments in fantasy settings.
The short answer is that it's the other way around, movie music developped from existing symphonic music.
The long answer goes back all the way to the early days of the film industry - and well before, as incidental music to theatre pieces (without speaking of opera) would have been a well-known phenomenon to audiences back then. Since reproducing sound on the same reel as the picture was a technically difficult process which did not get a workable solution until the 1930s, early films were as you know silent. It was thus common practice to hire out musicians to provide sound effects (with percussion instruments or other accessories), cover the sound of the film projectors, and eventually to heighten the audience's interest by playing thematically appropriate music. Many early cinemas thus had in-house orchestras.
When film producers eventually took the next step (and the financial means!) to hire out composers to produce original music specifically for their motion pictures, this was with established composers, such as Saint-Saëns (L'assassinat du Duc de Guise, op. 128, dating from 1908, is usually cited as among if not the very first music composed for a movie - see here for a rendition of the music with the - bar the missing footage of the final minutes - otherwise silent film). In Germany, Gottfried Huppertz and several composers at the beginning of the sound-motion picture period (Max Steiner, Erich Korngold; both of whom emigrated to the USA and played a major role in Hollywood) were influenced by the ideas of Wagner's operas. Having this symphonic background in their minds and the means (the in-house orchestras of major cinemas, or of the Hollywood film producers) to bring it out in film, these early examples proved influential in (pardon the pun!) setting the tone.
As the orchestral tradition of Western classical music continued (and continues) to evolve, other composers and well-known figures from varied backgrounds - other than as already mentioned Prokofiev, one could also list Copland (The Heiress, 1949) or Bernstein (On the Waterfront, 1954) - were in their time called on to provide music for the film industry, again making the distinction between "symphony" music and "movie" music quite small. Industry conventions, working methods and several other elements eventually led to the emergence of today's movie composer role, but the distance between this and the so-to-say traditional composer has never been very large, other than the specific purpose of the composed music. As the most prominent example, John William's long-standing collaborations with the blockbuster productions of Lucas and Spielberg all feature his decidedly symphonic music, full again of romantic Leitmotive and complex harmonies, along with more modern stylistic elements, making full use of the orchestral sound palette. The continued popularity of this symphonic style amongst moviegoers, younger composers and directors and producers, is ultimately the reason why symphonic music sounds "very much like a movie score" to you (or, in the other direction, why movie music will frequently also sound very much like symphonic music to those more familiar with the latter).
If it's of further interest - here is not the place to give an exhaustive bibliography on the subject, nor is that the question you ask - there are several academic works on the matter, including Kathryn Kalinak's Film Music: A Very Short Introduction (OUP, 2010). You can also look up the Grove's article on film music.
As has been mentioned enough now, the reason is that movies are following symphonies, not the other way around.
As a fun fact, the same goes for other media, like computer games: these used to not have symphonic music in the distant 8-bit and 16-bit era simply because the old machines were not capable of reproducing this kind of fidelity. Later, as the audio chips became ever more powerful, we had a middle ground where games could use MIDI and later realtime sample mixing to approach symphonic pieces quite successfully. Eventually, today, most expensive (AA/AAA) games have music scores that are symphonic - sometimes recorded by actual orchestras (famously, check out Austin Wintory's score to the game Journey, like this live recording behind-the-scenes).
Also it just so happens that adapting more current music (rock, metal etc.) to pure symphonic versions seems to be quite popular. Or, speaking of which, even mixtures like Symphonic Metal.
It seems that this kind of music is indeed timeless, and while there certainly are movies or games that do not use such scores (e.g., the original "Highlander" with its predominantly Glam-Rock score by Queen, or "Thor 3" with its hilarious Led Zeppelin intro, or any number of movies with mixed OSTs), most do. Maybe it's the capability of these instruments to at the same time stay in the background in a quite restrained manner, just filling out some unwanted quiet, but then also to really play a role as the foreground star (i.e., some scenes in "Interstellar" with the absolutely bombastic horns from Hans Zimmer). You can get both something that is almost invisible/inaudible; and something which is completely grating your nerves (the classic high tension strings, for example in the old Hitchcock movies). Symphonic music seems to be easily able to encompass all emotions quite successfully.
Have a look at the life and music of Erich Korngold. Trained in "classical" music, and wanting to be an heir to the likes of Mozart, he found his way out of Austria and in the direction of Hollywood music. I have heard one of his operas (a true opera, in the sense that classicists would understand) performed at the San Francisco Opera. But he made his money writing scores for movies, back in the era when movies were movies and had movie stars (in the sense that cinema buffs would understand). So, it is not surprising that his movie scores sounded a lot like a classical composer wrote them!
Related: If you watch Bugs Bunny and related cartoons, created in the 1940s, a lot of the background music (sometimes sung by the cartoon characters) was well-known classical music. This is partly because the music was out of copyright, and thus could be freely used, paying only the musicians rather than paying the composer.
Several answers have said how film music imitates symphony, but I don't think that's quite getting to the point of how this particular piece sounds like film music. What I'd list is
In short, what we have here is the kind of sounds that maximize dramatic effect without much that would distract from what's happening on screen.
Of course, many symphonic works use such dramatic elements, but for most of them these are rather employed for emphasis around more intricate melodies and intertwining of voices, which are harder to follow for the listener and therefore something film composers rather tend to avoid. As Prokofiev does in this piece.
