Beethoven's original music was comprised of eleven musical numbers interspersed throughout the play. Liszt uses three of these numbers for his fantasy. Liszt wrote three versions of this fantasy, for piano solo, for two pianos, and for piano and orchestra. It is the version for piano and orchestra that is heard on the video.
Liszt was one of the best sight-readers ever known. He could take a piece of music he had never seen and play it perfectly, in tempo, at sight. He could reduce orchestral scores to the essence of the music and play the most complicated music from sight. It was also said that the only time Liszt could play a piece of music and be faithful to what was printed on the page was the first time. After that, he began to change things in the score to suit him, at least with the new composers of the time. He was forever tinkering with other composer's music as well as his own. This musical tinkering no doubt lead to his many transcriptions, and lead to things like the Ruins Of Athens Fantasy. But it also must be remembered that piano versions of great works were sometimes what made the work well known. The expense of an orchestra has always been great, no less so in Liszt's time, and to be able to hear a new orchestral work was a luxury many listeners did not have. Liszt himself made Berlioz's Symphonie Fantastique more well known when he would play his piano solo version of it in recital. As it had been many years since Beethoven's original music had been heard, Liszt no doubt wanted to expose the listener to what he considered some of the best parts of it. Liszt was a man inspired by other composers music in many ways. The use of another composer's tunes can be a sign of respect, and with Liszt's known regard for Beethoven's music, he no doubt meant it as such.