Exported MP3 very loud

First, I'm curious which commercial CD you tried that turned out not to be normalized - this would be extremely unusual!

Anyhow, as to why normalization often means an increase in volume, consider that MuseScore can be used to produce scores for an entire orchestra with 100 musicians all playing "fff" dynamic, so the default volume level within MuseScore has to be set low enough that this would not distort. And yet another score might be a for a solo flute playing "ppp" the whole time, so that's going to be very quiet in comparison. The first score will sound basically the same after normalization, the second will get a lot louder , others will be somewhere in between.

Note that if you use no dynamics in your MuseScore file, the loudness of the overall resulting file will be quite constant. That means that after normalization your entire file will be perceived as "loud".
However the music ripped from a CD does have a dynamic range, some passages are more loud than others. Even when normalized, most of the song will sound "normal" and it only has some louder peaks when all instruments/the full band play all out at the same (short) time the drumset hits all its stuff at the same time.

Normalization only tries to ensure that all loudest peaks of your files end up at the same volume level.

Great! At the risk of confusing things further (but hopefully not):

It is indeed true that music with no dynamics will end up uniformly loud, and that most real music isn't that way, so only the "peaks" will be loud but the rest will be quieter in comparison. This in itself is something of a problem in the audio industry, which is why "compression" exists and is used in conjunction with normalization.

Consider:

Your goal in producing a song is usually for the volume to be similar to every other song out there, so your listener doesn't have to turn the volume up or down when your song comes on the radio or playlist (or have to endure the too-loud or too-soft result if he doesn't). Normalization is supposed to do that, by identifying the loudest passage - the "peak" - and increasing the overall volume of the track by whatever it takes so the peak is at the "normal" level - which is to say, your peak volume will be the same as everyone else's.

What often happens, though - particularly in classical, jazz, and other music outside the world of pop - is that there might be only one brief peak, and the rest of the music is much quieter in comparison. The result is that while your peak volume might be the same as the peak of other songs, most of your song is a lot quieter, and the overall perception of the listener will be that your song is quieter than everything else. The peaks might be the same, but since most of your song is much softer than the peak, he's still going to turn the volume up (and then potentially be annoyed when the peak comes and seems too loud).

The solution to this is compression - artificially lowering the peaks (or raising everything else) so the peaks aren't that much different from the rest of the song. After compression, then , the peaks will still be at "normal" level, but since they aren't that much higher than the rest of the song, the rest of your song won't seem so quiet in comparison.

This basically means that the loudest and softest passages in music that has been compressed in this fashion won't be all that different. The vast majority of popular music is verily heavily compressed, with the softest passages being only barely softer than the loudest. This is pretty much taken for granted as acceptable. Classical and jazz producers, on the other hand, typically struggle with this a lot more. On one hand they don't their music to seem much quieter than everyone else's, but on the other they want their actual dynamics to be honored. So they try to find a balance - how much compression to apply so the average volume isn't too soft, but the loudest passages will still be very noticeably louder. You'll see musicians and mastering engineers fighting about this all the time.

Anyhow, sounds like your song - since it has no dynamics whatsoever - was basically already complete compressed, so indeed the softest passages were as loud as the loudest, thus making it seem louder than music that is not so heavily compressed. Peak volumes were doubtless the same, but your softest passages were no quieter whereas in other music, the softer passages are at least somewhat quieter (more so for classical or jazz than pop usually).

Not sure if that information was helpful or interesting to you right now, but there it is for future reference :-)