When writing, you're told to 'show, not tell'. How exactly are you meant to do this?

Another example.

(Telling) I walked outside to grab the newspaper and saw a peculiar headline.

(Showing) I cleared the sand from my eyes, yawned and twisted the knob to the front door. It was a brisk and refreshing morning and the twilight of the sunrise gleamed across the freshly frosted grass. I took a deep breath, the crisp air was inviting, Johnny the newspaper boy had already started his route, a fresh Sunday Tribune was awaiting me on the doormat. Through the red plastic wrap I noticed something peculiar about the front page headline. I quickly pulled the paper from its wrap and there it was, the words that changed the world forever ... "...We Are Not Alone..." 

You "paint" with words. For example;

It was a cold rainy day.

Or,

The cold air caused the rain to bite on the skin.

See the difference?