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 If you can't understand what your partner is saying it might end up to be a huge hindrance to the effectiveness of your interpersonal English communication, so when trying to figure out how to improve spoken English, you should not neglect this area. The speech is too fast, or the accent is too strong - it is all a part of the same problem. So, again, you will need a step-by-step plan on solving the problem.

Figure out exactly what you don't understand

Figuring out the problem is already half of the solution, as they say.

Do you not understand the structure of sentences, or do you not understand plainly because you lack vocabulary? Or just you lack comprehension of the culture that makes the slang thrown at you to be alien language?

There is an example of a movie Fight Club, that I watched a long time ago in English, and I didn't understand a single thing in it. I turned up the subtitles and still had a hard time understanding it. Too slang-heavy, cool, bad-ass movie.

Fight Club has a deep meaning, as well, so there were a lot of metaphors and it was hard to catch their meanings, but anyway.

I ended up just not doing anything about it. But years later, I watched again some of it on YouTube and realized that I have improved a lot in comprehending the meaning. If I try to backtrack what exactly happened and what was conducive to my improvement, then it would probably boil down to these several points

Expose yourself to a variety of spoken and written English.

All I am doing is trying to backward engineer my results, so it is all subjective, but for the heck of it, it probably works.

You have to start listening a lot to a variety of English. And a lot. And that includes written English.

You have to start listening to academics lectures, to cartoons, to movies and, if you have an opportunity, to real people. I have to actually ask "Sorry, can you repeat yourself?" a lot lately, if I am talking to people with specific accents.

Very important - get used to the structure of sentences and grammar. They help to really understand some minute information nuances.

Do not use subtitles

All they end up doing is diverting your attention to them. You end up reading them, not listening to the actual speech being said. Try it for yourself and see your behavioral patterns. I bet you will read them more than you actually listen to speech. They might help at initial stage, but just don't get used to it.

I personally would turn them up sometimes, but usually I just turn them off. Expose your brain to too much simplicity and it will get lazy and end up not improving much.

Vocabulary

Some smart people say "just listen to English and you will guess the meaning of words automatically". Well, no. There was a reason my listening comprehension increased dramatically after learning GRE list, which has like 3500 weird words.

When you hear such words as "ameliorate", "abate", "denunciate" there are multiple ways a human can interpret them. As a matter of fact, I was once a believer that you will just "get the meaning of words", but when I realized my scores in GRE really suffered despite of this flawless strategy I started really thinking deeper.

Put just extra effort into learning some exact meanings and you will be rewarded. That's probably the core of how to improve spoken English.

Good luck with your studies on English language, folks.