In short, it's about two things:
Most cynically, money. Catalonia sends more money to Madrid than it gets back. If they are officially independent and sovereign, they get to keep that money, and Madrid loses that money (and any legal way of getting their hands on it).
More generally, overall concept of sovereignty.
This includes things like control of one's destiny, control over one's territory, foreign policy, and, lest we forget, control over one's money/budget.
Additionally, the question seems to be based on an invalid premise that sovereignty has anything to do with laws of original country. It does not, in any meaningful way.
Sovereignty is ONLY about the actual, practical ability to exercise sovereign control.
That can be independent (you are able and willing to repel any forces another country sends to take the control from you) or international (there exists a sufficient amount of other countries that not simply recognize your sovereignty, but are willing to back up that recognition in a meaningful way, either peacefully - e.g. through UN resolutions - but ultimately, through the use of force.
Ultimately, like any politics involving the use of force internationally, it boils down to creating actual conditions such that the original country is no longer either able or willing to wrestle back control - either by being defeated by force or by its populace losing will/interest in such things (the latter happened with most colonial powers in 20th century).
As an example, look at some other independence cases:
There was no provision in Serbian constitution for Kosovo to separate. But Kosovo was able to first wage successful armed struggle to push Serbians out; then win enough international support (Western Europe, then US) to prevent Serbia (and Russia which supported them) from wrestling the control back.
Ossetia, Abkhazia, and Transdnestria (and seems to roll similar way in near future for Donetsk etc...) - Same exact story with Russia replacing USA as external force guarantor. None of them happened in accordance to the laws or the will of original country.
United States of America - waged the war of Independence against UK, and also got some international support, most notably from France. Clearly not according to the laws of UK at the time.