Writing Tips: Fleshing out the Dialogue

Writing dialogue is important. It often seems straightforward. Sometimes you might struggle with it; at other times, it flows from your brain through your fingertips and onto the page. You sit back and smile at it and be proud!
Then you take it to somebody else to read. And their first reaction is, "What is actually going on in this scene?"
That's when you realize that you've written talking heads.
First off, don't panic. Everyone does this. You write out the dialogue and forget about the people talking. But the solution is not as hard as it appears.
First get rid of the word said. "He said, she said" is straightforward, but blander than plain spaghetti noodles if you use it too often. Look up some synonyms for said, and find the ones that fit the scene best. (It also works into the Show, don't Tell.) Contrast this: "You could have told me earlier!" he said angrily with this: "You should have told me earlier!" he snarled.  One word less, and we know instantly that he is quite angry, and hear his tone of voice. Or compare "We've got everything we need," she said happily with "We've got everything we need," she chirped. The word chirped in this sense tells us that she is happy, lets us hear her piping  voice, and gives us a glimpse into her personality.
Another way of fleshing out the dialogue is to show facial expressions and movements. It's possible to get rid of the said synonyms as well, by putting action in front of or behind the sentence. For example:
"It's well past time you got rid of this old chair." John ran his finger down the tattered upholstery, picking at the loose threads. 
Maggie sniffed, tossing her hair over her shoulder. " I'll thank you to leave that alone! It was Grandma's favorite chair."
John rolled his eyes, but leaned back into the chair and stopped tugging the threads. "As you wish, madame."
Now, it's possible to abuse this technique: you don't have to put an elaborate motion in front of every sentence. But using it helps paint a clear picture of what's going on, and may give a better idea of what your characters are like. In the above example, we can tell not only that John dislikes the ratty old chair, but that he's a bit sarcastic and doesn't pick fights. And Maggie likes her old Grandma, is fond of the chair, and is either haughty or putting him on. We know this from the words and the actions. And we know this in only three sentences.
A simple motion or expression will have the same effect. Smiling, scowling, blinking, turning away, rubbing their head, or biting their lip gives a quick indication of the character's mood, and how they said their line. You can also use random actions to give the illusion that this is a living person, instead of just words: sneezing, coughing at stirred dust, stumbling, yawning, or accidentally dropping something.
The best way to figure out how to do this and make it look natural is to watch people--not necessarily on television, either, since acting is not always natural. Watch people in real life talking, and see how they move and act with their words. Note the various quirks people have, and come up with similar ones for your characters.
Not only does this make for a clearer scene, but also for more engaging characters--and stories people will want to read.
And remember: that's your actual goal.