Firstly, variety in sentence length is important to keep the prose from growing repetitive. While reading, most people probably aren't aware that the writing has a rhythm, but when sentence lengths and structures are the same over and over again, the writing is not dynamic and readers will grow bored and possibly not know why. Most writers tend to have an ideal sentence length they tend to fall back on, so it's important to be conscious of it. If you usually write 7 word sentences, break the mould and write a few 20 word ones,
Sentence structure is another place you need variety to keep the prose lively. It's easy to fall into a particular sequence of subject, verb, noun. Eg. Ted crossed the room. Fine. It gets the message across. But if you follow that sentence with one like 'He switched on the light', it gets repetitive. Try changing the order of the words in some sentences, putting the verb first, or the noun. You'll be amazed at how much more zing your prose has once you start being aware of your sentence structure.
Variety in word choice should go without saying. The English language is full of wonderful words, but many of us stick to a limited vocabulary, even in our writing. I know that in my own work, I overuse the word 'look'. In revision, I go through and search for all my uses of the word and change every second or third one to another word like gaze, stare, eye, peer, peek, regard, study, or whichever suits the scene. You want to limit word repetitions within pages and chapters to keep the writing tight.
These are just a few things that might help make your writing sing from the page. Variety really is the spice of life, isn't it?