All audio you record has a dynamic range - the range between the quietest and loudest sounds you made and stored in your file. When you record, it's very unlikely that you will get the levels in the optimum place, so that they sound loud enough without overloading. What normalizing does is shift the level (usually up) so that the highest peaks reach just below the distortion point - 0dB. This process doesn't alter the recording in any way - it just shifts the whole thing up or down, to a point you can select. It's usual to set this to a dB figure - like -1dB.
There is more than one good reason for doing this. Firstly, it means that you can hear your recording at a sensible level without having to crank the volume up on whatever you are monitoring on. Secondly, it means that you will be able to see the waveform a lot more clearly - it's very difficult to work with a very small waveform. Thirdly, any effects that do dynamics processing (most of the stuff in 'Amplitude and Compression' in Effects) are designed to work correctly with signals that have been normalized before processing, and if you don't do this, they simply won't work properly.
Typical use case: When you record using a microphone, or from any source you don't have control of, it's normal to leave what is known as 'headroom' - you deliberately don't try to record up to the peak, which leaves room for people to suddenly get louder than you thought they would. For example, with most music recording, I leave at least 12dB. This doesn't compromise that system's dynamic range, and it means that I don't get caught out with peaks. But after the recording I want the recording not to peak at -12dB, but at -1dB. And that's achieved simply by normalizing to that level.
There is more to it as well in terms of implications, but for starters that should do...
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