Since sexual intercourse is the defining act of marriage, it's helpful to look at what the Catholic Church says about marriage. Traditionally, Catholic theologians have identified three purposes of marriage, which I present here in no particular order:
- Mutual help of the spouses (which I believe has developed more recently into the notion of the union of the spouses)
- Procreation and education of children
- Relief of concupiscence
Generally speaking, I think the prohibition of artificial birth control follows from the principle that married couples should use their sexuality in a way that always respects all three of the basic purposes of marriage. But the use of artificial birth control implies the ordering of a particular act of sexual intercourse in such a way that it is intrinsically incapable of procreation. Before he was Pope John Paul II, Karol Wojtyla argued further, in Love and Responsibility: that choosing to withhold one's fertility in sex is also to work against the unity of the spouses.
All told then, the Catholic philosophical view could be summed up thus: artificial birth control makes marriage one-dimensional, by deliberately limiting its defining act to the service of the least noble of the three purposes of marriage.
Plenty of objections arise on the way from the principle to the conclusion, of course -- particularly in the articulation of how abstinence-based methods can be morally legitimate while artificial methods and sterilization cannot -- but I'm not sure if this is the right venue for considering and responding to all of those.
One thing I'd like to try to clear up, though, if I might. In your question you said:
However, planning to wait a little bit here and there seems reasonable enough.
Of course, and this is precisely what the Catholic Church teaches, at least as I understand it. The clearest official statement of this point that I've seen is Gaudium et Spes paragraph 50.
The Church certainly does not require its faithful to have as many children as they're capable of, nor does it require every act of sexual intercourse to be aimed at procreation -- just to set aside two very common misunderstandings. One has to bear in mind that one of the purposes of marriage, in Catholic thought, is the "procreation and education [or bringing-up] of children". Every married couple must discern for themselves the right balance between fruitfulness and their responsibility for the children they already have.
The main thing is that any "waiting a little bit here and there" be achieved by morally acceptable means. This is of course what the comments on your question were getting at.