How can cats' claw grow sharper without replacing the claw?
The horn has a specific structure which is different from the round edge that we see on our fingers. It splinters and cracks into flakes, so it is more similar to a flint than a chalk. 
The outside of the cone dries out and flakes and is scratched off by the cat, while the inside of the cone is the newer horn that grows towards the apex as a new blade edge.



The image is from this research. If you search "sharp" you will find astonishing vetinary descriptions quoted. Also see page 641 for the conclusion of the research. He sais that it seems to have evolved in the aim of keeping the feline claws sharp.

If someone has the technical ability to rewrite the researcher's descriptions into everyday words, you would be a first.

Vetinary research quote:

The horn-generating living blade epidermis that is
located most centrally on the slanted face of the curved
ledge is continuous with the parietal epidermis (Fig. 9b),
so that the blade epidermal cells generated here lie next
to the parietal epidermal cells. Therefore, the cornifying
parietal epidermal cells are deflected and entrained apically
as the blade horn is pushed apically by the proliferating living
blade epidermis. The direction in which the blade horn is
pushed forward can also be inferred from the arrangement
of the ridges visible on the dermo–epidermal interface of
the parietal horn (Fig. 7). The living blade epidermis that
is located at the outer edge of the face of the curved ledge
is continuous with the living epidermis of the proximal
cone horn (Fig. 9b). These lateralmost blade epidermal
cells, therefore, form a layer that lies directly underneath
the deepest layer of proximal cone horn, so that the
boundary between the blade horn and proximal cone horn
is indistinct. Hence, the layers of horn that are generated
in the dorso-proximal bend of the curved ledge underneath the combined horn layer of the external coronary
and proximal cone horn cover the sides of the apical cone
horn and are designated as part of the blade horn. As
the superficial coronary horn on the dorsal side of the
cornified claw sheath is worn off through normal use, the
superficial layers of the blade horn are exposed and are
shown to hug the sides of the apical cone horn (Fig. 13a).
Remnants of these blade horn layers are discernible near
the very tip of the cornified claw sheath, which consists
mostly of apical cone horn, because the rest of the coronary
horn, including most of the blade horn, has been worn off
(Fig. 13b,c).
The remaining soft-cornified horn segments of the
cornified claw sheath are found in places where the
hard-cornified horn divisions need to be supported and
cushioned.
The soft-cornified terminal horn is a segment of the
third major horn division (i.e. the Hyponychium; see Table 2).
It is generated by the living terminal epidermis, which is
continuous caudally with the parietal living epidermis and
is supported by the dermis that covers the medial, lateral
and palmar sides of the tip of the bony unguicular process
(Figs 3a, 4a,c, 7, 9b, 11a, 12a–j). It supports the apical cone
and crest horn on its dorso-distal side (Figs 3a, 4a,c, 7, 9b,
13b) and the blade horn on its lateral and medial sides
(Figs 4b, 8, 9b–d, 13a,b). By filling the space created by
these converging segments of hard-cornified internal
coronary horn, it is shaped into an elongated, pointed
cone and extends proximally between the blade and
sole horn (Figs 12c,d,f,h,j, 13a,b). Because the terminal
horn is soft-cornified, it can adjust its shape to the
changing conformation of the space it fills in the course of
the deformations of the cornified claw sheath during its
normal use.
The friable soft-cornified sole horn is generated by the
living sole epidermis, which is supported by the dermis
that covers the proximo-palmar side of the bony unguicular
process (Figs 3a, 4, 7, 8, 9b–e, 12c,d, 13a,b). Along its medial
and lateral sides, it is embraced by the terminal and parietal
horn (Figs 1b, 4b, 8, 9c,d, 13a,b). The sole horn, therefore, fills
the space between the paired cutting edges of the cornified
claw sheath. It is worn off through natural wear and tear,
as well as through repeated compression and release, as
the cornified claw sheaths are deformed when they are
subjected to tensile forces during climbing and prey capture.