Native French speakers do like to talk at quite a clip! It's also true that to non-francophone ears, native French speakers can seem to almost drop words or parts of words when they're speaking quickly. All of this can make it quite difficult for learners to understand what's being said!
As you say yourself, the main advice here will indeed be "practice, practice, practice" - and it does absolutely get easier! It's about training your ear to hear the things that are usually glossed over, and about training your brain to keep up with rapid-fire French!
It's not a very high-tech solution, but something that may help you to hear those words that get dropped is to keep your cursor or finger over the text being read as you listen to the audio. Move your cursor/finger along the text as the speaker speaks - as you keep time with them and listen carefully, you should be able to start to catch the bits of speech that are dropped, or at least see where you're missing things. Wearing headphones while you do this will make this easier, as it will help you to isolate the sound of the speaker's voice.
As for comprehension in general, it is absolutely useful to keep an ear out for certain keywords that you know should come up in the sentence. Another thing is just to accept that you may not understand absolutely everything a native speaker says at the start. It's like when you start reading more complicated texts: you aren't going to know every word on the page either. The key thing is to try to get the gist, and to catch the words that at least give you the main topic of the sentence. Then, as you get more practice, you will understand more and more of the sentence.
I hope that this was helpful! Perhaps someone else on the forum will be able to offer their own advice as well.
Bon courage !