This is a bit subtle. I'm guessing "case of knowledge justified through reason alone" is pointing to the Kantian conception of mathematics as synthetic a priori knowledge. Frege really wanted to be a Kantian and expressed great admiration for Kant, whom had come up with the distinction between synthetic a priori and analytic a priori knowledge. But at the same time, Frege thought Kant was wrong about arithmetic being synthetic a priori and instead argued that arithmetic is analytic a priori.
Basically Kant had argued that the way we grasped 2 + 3 = 5 is counting up some things, so the result is synthetic. (He wrote something to the effect that no matter how much you stare at the concept of two and at the concept of three separately, the concept of five just doesn't pop up.) An analytic result for Kant was something like saying that the sum is greater than the part; basically something that follows from definitions. Frege instead argued that arithmetic could be axiomatized so it could be exactly like that.
(There is actually a bit more to be said here, as Frege tried to defend Kant's idea that geometry was synthetic, even though non-Euclidean geometry had been invented/discovered by that time. Frege argued that non-Euclidean geometry was non-intuitive, unlike arithmetic. Why geometry and arithmetic feature in this debate? Those were basically the only branches of math known to Kant.)
Gilles' book "Frege, Dedeking, Peano, and the Foundations of Arithmetic" is a really good philosophical read on this.
I suppose the way Payne presents this is because he thinks (if not argues explicitly--I don't have the book on hand to check) is that Kant is considered a rationalist.
You are however quite right to be riled by that because rationalism is often contrasted with empiricism. A prime exponent of empiricism at the time was J.S. Mill, who argued that mathematics was ultimately an inductive affair; we discover mathematical truths by examining nature basically. (At the very least Mill argued that axiom are discovered that way. Mill also argued that all results in arithmetic are empirical since we can basically check them that way.) Frege was also utterly opposed to Mill's conception of mathematics; unlike his expressed admiration for Kant, Frege could barely contain his disdain toward Mill. Which may seem a bit odd given that there doesn't appear to be a lot of difference between what Kant and Mill said about arithmetic. Frege's main beef with Mill was the latter held that numbers always come up from some application (i.e. that 1+2=3 always meant something like one apple plus two apples equals thee apples) whereas Frege (like Kant basically) was a Platonist, holding that the numbers that we operate on in arithmetic are ideal concepts.
Contra Frege, Gilles argues that from a modern perspective it's obvious that Frege was wrong about (non-Euclidean) geometry and effectively that the Kantian vision of mathematics as synthetic a priori knowledge had been blown up by Frege's work on arithmetic and the discovery of non-Euclidean geometry. It seems to me it's this erosion/destruction of the synthetic-analytic distinction with respect to mathematics as what Payne phrases as "Logic doesn’t constitute knowledge of the world, it is merely a tool for organizing knowledge [...] Mathematics had long served as the rationalist’s paradigm case of knowledge justified through reason alone." Payne also says "Logical Positivism can be understood as Empiricism", although Frege would have probably disagreed with that bit (if he ever identified as a "positivist", which I don't think he did).
There's a SEP page that discusses "logical empiricism" and it basically picks up from where Frege left the matters:
Carnap had not only studied with Frege, but like many of the logical empiricists he had started out as a neo-Kantian as well. So especially in view of Russell’s relatively more successful attempt at reducing mathematics to logic, it was perhaps natural that Carnap would consider both mathematics and logic as analytic. [...] But from fairly early on there was widespread agreement among the logical empiricists that there was no synthetic a priori, and that logic and mathematics and perhaps much else that seemed impervious to empirical disconfirmation should be thought of as analytic. The point of drawing the analytic-synthetic distinction, then, is not to divide the body of scientific truths or to divide philosophy from science, but to show how to integrate them into a natural scientific whole. Along the way the distinction clarifies which inferences are to be taken as legitimate and which are not. If, as Carnap and Neurath were, you are impressed by Duhemian arguments to the effect that generally claims must be combined in order to test them, the analytic-synthetic distinction allows you to clarify which combinations of claims are testable.
As another book notes somewhat hagiographically towards Frege, I'd say, but also noting the substantial difference I mentioned earlier between Frege and positivists:
Among the first appropriators of Frege’s work were the logical positivists who dominated philosophy in the period between the 1920s and 1950. Although
among Frege’s three original promoters, Russell and Wittgenstein have greater
individual philosophical stature than Carnap, Carnap was part of a movement—logical positivism—that had the broadest influence among those who
first made use of Frege’s work. [...]
inherited from Frege an interest in meaning as expressed in language. Both
Frege and his positivist successors were concerned to understand language
and meaning, because they saw such understanding as a new and promising
route to understanding the nature of human knowledge. Frege and the positivists took human knowledge to be best exemplified by scientific knowledge.
(As we shall see, however, they understood this point in very different ways.)
This approach to philosophy through a consideration of the nature of human
knowledge is, of course, traditional. It had dominated the subject from
Descartes through Kant. Kant was a source of inspiration for both Frege
and his positivist successors.
In broadest terms, Frege added two things to this tradition. He added a
concentration on language in the expression of knowledge. And he added
a recognition of the power of logic to illuminate the structure of language and
its contribution to the expression of knowledge. Frege is responsible for
establishing and developing modern logic—a huge achievement. [...]
Frege’s positivist successors used Frege’s innovations. Yet they did so
against a background of philosophical attitudes that Frege did not share. They
differed with him, in absolutely fundamental ways, about both meaning and
Although the positivists took up Frege’s focus on language, they appended
to it an ideology of exclusivism. The only meaning that might be of any
cognitive value was, for them, scientific meaning. It is this ideology that led
positivism to be aggressively deflationary about those aspects of philosophy
and culture that it could not assimilate to a scientific paradigm.
[...] Whereas Frege’s successors saw themselves as overturning traditional
philosophy, Frege saw himself as continuing a philosophical tradition.
Whereas his positivist successors applied reductionistic or deflationary attitudes to nearly all philosophical problems, Frege confined his reductionism to the attempt to reduce the mathematics of number to logic. He shows no
special inclination to hold that reduction is a good method in philosophy
Likewise Russell influenced positivism, but it's been disputed whether he substantially subscribed to that view himself. (More could be said on this, but I'm a bit short of time now.)