Writing lyrics is like anything else. You have to practice and practice at it to get good. Some people might be naturally better at it, but that doesn't mean that you can't get that good.
A good way to practice is to start reading and writing poetry, because in essence lyrics are just poetry. Study the forms and structures of poetry and you will be one step closer to writing lyrics.
Also lyrics don't have to make sense. Some lyrics are abstract, almost unintelligible, and meaningless to many. Some people write lyrics just because they have nice form and the words sound good together, but together they don't actually make sense as a whole.
I would also listen very closely to the lyrics of your favorite artists. You will be able to follow the story of some songs while others will make no sense. One particular artists that stands out in my mind is Bon Iver. Some of his lyrics make a little sense but otherwise I don't understand what he is singing about. While on the flip-side an artist like Carrie Underwood tells a very clear story with a lot of her songs.
When I was first writing songs I was terrible and my lyrics were cheesy, cliche, and just plain boring. Now that I have been writing for several years my lyrics are starting to take on actual meaning and have some depth.
It takes time. Don't give up and keep writing and you will just get better and better. Here are several books you could purchase (perhaps get at your local library) that might help you.
Writing Better Lyrics
The Craft of Lyric Writing
Popular Lyric Writing 10 Steps to Effective Storytelling
Good luck and keep writing!