Why do spacecraft need heat shields coming back to Earth but not leaving?
First, I should point out that spacecraft do use heat shields when leaving Earth - however, you are right that much more heat is generated when they come back into the atmosphere.
To understand why, you first need to understand a little bit about atmospheric drag. Drag is the resistance of the atmosphere to the fact that you're trying to plow something through it. It slows you down and turns some of your energy into heat.
The amount of drag experienced by an object depends on several factors: the density of the atmosphere (the denser the atmosphere, the more drag it produces), the speed of the object (the faster it is, the more drag it feels) and the cross-sectional area and shape of the object perpendicular to its direction of motion (think of a parachute, which has a big surface area in order to produce a large amount of drag and slow you down, vs. a race car, which has a nice sleek shape to minimize the amount of drag).
Looking at the above factors, we can see that at a given point in the atmosphere, the only way the drag will be different is if you change the spacecraft's speed or its cross-sectional area and shape. It turns out that both these factors are different on the way up than on the way down.
Atmospheric drag would be a major problem on the way up if you had to give a spacecraft all the speed it needs to get into orbit right at the moment of launch. Fortunately, if you've ever seen a Shuttle (or other spacecraft) launch you know this is not what happens - instead, the Shuttle rises slowly at first and eventually gains speed as it continues to burn fuel and fire its engines on the way up (for many more details on the process of getting into orbit, see this page from the HowStuffWorks website or this one from The Launch Zone).
As you get higher in the atmosphere, the density drops off rapidly (exponentially, in fact), so once you are high up you can increase your speed significantly and still not create too much drag. Engineers talk about the point of maximum q - this is where the atmospheric drag has reached its maximum value. This occurs around one minute after launch and at a height of several kilometers - after this point, the Shuttle engines are put on maximum throttle (though the Shuttle doesn't actually reach its maximum speed until much higher in the atmosphere).
Okay, so what about coming back to Earth? When in orbit, the Shuttle is moving very fast - nearly 8 kilometers per second (over 17,000 miles per hour). In order to bring it back to Earth safely, you need to slow it down considerably - if you've ever seen the Shuttle land, you know it touches down much like an airplane does, and therefore is going at a relatively slow speed.
One way to slow the Shuttle down is by firing its rockets - a similar procedure to the one used to launch it. However, this requires an enormous amount of fuel (as you can see when you watch a launch), and it would be prohibitive to lug all that extra fuel up into space with you just so you can use it on the way down!
A more efficient method is to do a relatively small burn with the main engines (which brings the Shuttle down to a lower orbit where it is in contact with more of the atmosphere) and then to allow atmospheric drag to do the rest of the work for you - in other words, atmospheric drag is intentionally used to slow down the Shuttle, so you want to generate a lot of heat on the way back to Earth! You may have noticed that on the way up, the Shuttle goes "pointy end" first - its shape is very aerodynamic (just like a race car) in order to minimize drag. On the way down, though, it hits the atmosphere with its big black belly and descends for a while in this "belly flop" position - it now presents a much less aerodynamic shape to the atmosphere (just like a parachute) which is used to slow it down.
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