Better than just reading the letter is to listen to the musical change in Beethoven's works.
Beethoven's music is usually divided into 3 periods. The first is really classical sounding, like late Mozart. In fact, Beethoven was a fan of Mozart and some sources say that Mozart appreciated Beethoven's compositions too. Go listen to string quartet no. 1, or symphony no. 1 for a bit. StructureStructured, restrained, even elegant. There's definitely emotionality in there especially compared to peak Classical era like Haydn. Those guys may as well have shit marble, as Mozart put it. Check out the pathetique sonata for something about as emotive as Beethoven got at the time.
Then came the diagnosis of progressive deafness. This letter is the best insight into Beethoven's mind at the time, and "existential depression with suicidal ideation" seems like the right description to me. He stopped composing for several months.
When he returned to composing in 1803, it's a totally different sound. This is the wild, emotional Beethoven we generally think of. Waldstein sonata, symphony 5, string quartet no. 9 are good examples here. It's Rage Against The Machine before electric instruments were invented.
If you're interested in the third period, read on.
The transition to third period is more gradual, around 1815. There isn't a hard biographical line in the sand like the Heiligenstadt testament, but his relationship with his nephew/ward Karl got awful, his love life fell apart, and his deafness took a sharp turn for the worse, all around the same time. His old tricks to keep composing, like sawing off the legs of his piano so he could feel it in the floor (poor downstairs neighbors!) were not effective anymore, and he was reduced to mostly communicating on a chalk tablet.
His music turned inwards, became more introspective. Themes like life and death and afterlife, godliness and debasement, retribution and redemption flood the work. It also gets much more complicated harmonically. He does a lot of contrasts, between the grandiose and the intimate, for example. Symphony 9, particularly the Ode To Joy, is quintessential. Also check out string quartet op.131, the Hammerklavier piano sonata, and the missa solemnis in D.
Now go out and impress your friends at dinner parties with your deep knowledge of Beethoven! Tune in next time for Bach: the original "ba-rock" star.