In my first year of college, my teacher would say "YOU. ARE. PLAYING. AS. IF. YOU. TALKED. LIKE. THIS." Meaning, that I wasn't putting any phrasing into the piece.
As others have said, it's analogous to the way that you change emphasis on syllables when you talk. I'll use the (apropos) word "frustrating" as an example. First, there is an accent on the first syllable: "FRUS-tra-ting." (We'll assume that you're not one of the Brits who says "frus-TRA-ting" as some do.) That doesn't change. But what does change is how strongly you emphasize it. "Yes, I can see what you mean, that's frustrating" might have one emphasis, while "That's SO FRUStrating!" might have a stronger emphasis to convey a greater degree of frustration. Then again, you could say "That's SOOO Frustrating" again conveying a different shade of meaning. That's basically what phrasing is.
It's the same with music. You have the meter, which needs to come out (ONE-two-Three-four ONE-two-Three-four), and you also have subtle (and sometimes not-so-subtle) variations in volume and duration (yes, it is both) that are a part of phrasing.
If you want to study phrasing, have a look at Chopin's Prelude No. 7, the "easy" one. Start with Argerich's version in the link that alephzero has posted above (it's at about 7:40). Then, go in and look at some others. There are several that have the music running along with them, and you can look at the phrase markings (the curvy lines over groups of notes) to get some idea of Chopin's intent.
This very simple piece is often thought of as one of the hardest to perform, simply because people try so hard to "phrase" it that they wind up overcomplicating things to the point of ridiculousness. You should find several examples of this. (Some of the performances remind me of the overemphasized gestures that actors were fond of making in the silent movies, that seem so silly now.) You can use Argerich's verson as a standard of clean, simple phrasing for comparison. Since you're the analyzing type, look at the slight variations in volume, the slight variations in legato, and the slight variations in meter that she uses. And, look at how these things are overdone in versions that seem over the top to you.
And then keep in mind that phrasing is ultimately intuitive. Just as there's a natural way to talk, there's a natural way to sing, and by extension to play an instrument. If you have the idea that a "good speech" applies all the best techniques of public speaking, and work on the delivery of every phrase, all the while being nervous about whether you really have anything to say, your speech will sound contrived and artificial no matter how profound the content. So it is with musical phrasing. If you try to phrase a piece by analyzing shadings, dynamics, articulation and so on, while still being nervous about sharing your heart with the world, then no matter how good your analysis is, and how well you execute it, it will still come across as artificial and contrived.
Let the music be a means to share your heart with the world, and phrasing will take care of itself. It's a natural thing.