How come masturbation is sin?

The Catechism treats this directly in section 2352 (see also 2354, which concerns pornography), but I will try to expand the answer a little bit. Bear in mind that this question treats some adult themes!

Let’s remember, in the first place, that the Church looks at our sexuality and the sexual act as something very holy and worthy of great dignity and respect. Sex is the highest and holiest expression of married love. Therefore it should not be treated as something “dirty,” but that requires us to look at it with pure hearts and pure minds.

There are two ends (that is, purposes) to the sexual act: the unitive end and the procreative end. In other words, the sexual act is meant to accomplish two very beautiful and holy objectives: bring a couple closer together in married love, and bring children into the world.

We can look at whether or not these two purposes of sex are being accomplished by looking at whether or not it is truly expressing the reality of sexual love, which is only possible in marriage–and married love is free, total, faithful, and fruitful.

• Free, because the couple is giving themselves to each other freely–they have chosen to make a gift of themselves. It’s not compulsory, involuntary, uncaring, or taken by force. Love requires decision.
• Total, or full, because the couple is not holding anything back from one another–they are not limiting themselves. This requires mutual trust and selflessness.
• Faithful, because their love is an expression of their commitment to one another. It can’t be bought, and it can’t be swapped out for another person.
• Fruitful, because it is life-giving. That does not mean that every sexual act between a couple leads to conception, but there is always an openness to life. It is also an expression of their spiritual fruitfulness, because they are giving of themselves to one another.

All sexual sins go against these qualities (“free, total, faithful, and fruitful”) in some way, preventing the two purposes of sex (union and procreation) from being accomplished. For example, adultery is a sin (and therefore not an expression of married love) because it is unfaithful–the people engaged in the sexual act are not actually expressing any faithfulness to one another.

In the case of masturbation, the sexual action is actually missing all of the qualities of sexual love. In a particular way, masturbation is not total or fruitful. There is no total gift of self in masturbation–in fact, there’s no gift at all; instead, it’s one person deriving personal sexual pleasure. It’s also fruitless–there’s no spiritual or biological possibility of life in masturbation.

From this we can see that masturbation doesn’t accomplish either of the purposes of the sexual act either. Neither union nor procreation is possible, because there’s nobody else involved–it’s a self-serving act. No gift of self, no expression of love.

It should be noted, though, that many people (men and women) who commit this sin are trapped in a habitual behavior. Where pornography is involved, this often leads to addiction. Compulsive masturbation and pornography addiction keeps a person trapped in a cycle of sin–and recent scientific research has shown that this cycle is not just spiritually harmful, but also psychologically and even physically harmful as well.
The good news is that if you do struggle with these sins, the Church wants you to be free, and there are many resources out there to help you! Consider checking out Chastity Project, Fortify, Exodus 90, Covenant Eyes, and speakers like Matt Fradd, Jason and Crystalina Evert, and Fr. Sean Kilcawley. They will also have a lot of additional resources to help you.