Short Answer
A thousand years ago, a person was readily culpable or not for their choices. Today, culpability is not so clear since we participate in highly complex societies built on highly complex ideas. In this way, Camus is calling attention to one's relationship to society and the world, as existentialists (he claims he's not) are prone to do. "Crimes of passion" are immoral decisions to harm based on narrow, emotional, and immediate impulses, but there is another class of immorality, those of "crimes of logic" such as when the CEO of a company makes a decision that has an immoral outcome, but is sanctioned under law and the folkways and mores of a complex society, particularly one built on ideologies. He's using a technicality of law as the basis of an extended metaphor to talk about how the very notion of guilt itself turns on ideology.
Long Answer
From the introduction:
There are crimes of passion and crimes of logic. The boundary between them is not clearly defined. But
the Penal Code makes the convenient distinction of premeditation. We are living in the era of
premeditation and the perfect crime. Our criminals are no longer helpless children who could plead love
as their excuse. On the contrary, they are adults and they have a perfect alibi: philosophy, which can be
used for any purpose—even for transforming murderers into judges.
Literally, the difference between a "crime of logic" and a "crime of passion" is a characterization of whether not the occurrence of a crime is premeditated or impulsive. It is an attempt to examine mens rea to classify the nature of the intention, if any at all. Sometimes, homicide, for instance, can be a result of negligence. For example, one can plan to assassinate a political leader and take many calm, collected acts to achieve the goal, or one might simply be enraged and under the duress of emotion accidentally kill someone. Simply, there are different types of homicide. Courts and jurists of all sorts generally recognize these distinctions in their taxonomy of crime and sentencing. The notion of intention is important to existentialists and absurdists because of the notion of being responsible for one's life, particularly after an occurrence of existential crisis.
Metaphorically, what Camus is doing is pointing out that not all crimes are seen as such since philosophy allows the disingenuous use of reason to justify immoral acts. Remember, Camus resisted the Nazis, and had seen first hand how legal and immoral needn't coincide. But one doesn't have to live under the NSDAP to find instances of where logic and words are used to obfuscate immoral acts. Camus had a fascination with the absurd, and it is certainly absurd that many virtuous people are imprisoned for "crimes" while many immoral scheming people are handed a pass because of how law is constructed. It's not only a question of how immoral people calculate to avoid punishment, but how law itself can be flawed. This is a theme of the short story Billy Budd by Melville.
Melville's dramatic presentation of the contradiction between the requirements of the law and the needs of humanity made the novella an iconic text in the field of law and literature. Earlier readers viewed Captain Vere as good man trapped by bad law.
In the second paragraph, Camus speaks more to law:
But as soon as a man,
through lack of character, takes refuge in doctrine, as soon as crime reasons about itself, it multiplies like
reason itself and assumes all the aspects of the syllogism. Once crime was as solitary as a cry of protest;
now it is as universal as science. Yesterday it was put on trial; today it determines the law.
But yet, there's another dimension, the rationalization of activity to avoid culpability. What is this "man who takes refuge in doctrine"? He is one who does not reflect in his tacit participation in social activity that has immoral outcomes. In other words, there are many people who are "just following orders" who choose to "go along, get along" without the slightest reflection that not only their participation in society might be negligent, it might be criminal even if not recognized as such. Then, he goes on to explain that "logical crime" is actually a metaphor.
This is not the place for indignation. The purpose of this essay is once again to face the reality of the
present, which is logical crime, and to examine meticulously the arguments by which it is justified...
So, clearly, he's using this distinction in the philosophy of law. He goes on to explain it further on:
In the age of ideologies, we must examine our position in relation to murder. If murder has
rational foundations, then our period and we ourselves are rationally consequent. If it has no rational
foundations, then we are insane and there is no alternative but to find some justification or to avert our
faces.
And there we have it. In an "age of ideologies", which is a stark contrast to the emotional, barbaric crimes of yesteryear, immorality and true criminality spring not from the primal human condition, but rather the impersonal constructions of ideas and societies organized around them. This raises the specter of whether or not we are guilty merely by existence in relation to our ideas and society.
Our purpose is to find out whether innocence, the moment it becomes involved in action, can avoid
committing murder. We can act only in terms of our own time, among the people who surround us. We
shall know nothing until we know whether we have the right to kill our fellow men, or the right to let
them be killed.
What relevance to us? It is one of guilt.
The philosopher Martin Buber underlined the difference between the Freudian notion of guilt, based on internal conflicts, and existential guilt, based on actual harm done to others
Did Hitler's people, ignorant of the Final Solution, bear some measure of responsibility for it given their endorsement of his government? Do, you, for that measure, bear some guilt for the death of wildlife, if by using electricity to power your computer, you slowly poison your community with mercury from the combustion of coal? Many would simply say no, but then to "prove" it, they would likely offer an explanation of how they are absolved of guilt. It's not illegal to use a computer. Everyone is doing it. The rights of an individual are more important than those of the government. In short, they would invoke philosophy and commit a "crime of logic".