Why is the mosquito a definitive host in malaria?
Plasmodium falciparum (the main causative agent of malaria) and other Plasmodium species have a very complex life cycle, with stages in the female host Anopheles mosquito, in the human liver, and in the human circulatory system, where it primarily resides in the erythrocytes (red blood cells, or RBCs):



During a blood meal, a malaria-infected mosquito injects sporozoites in its saliva into the human host (1). Sporozoites travel through the circulatory system and first infect hepatocytes (liver cells) (2, stage A) and mature into schizonts (3), which rupture and release merozoites (4). After this initial replication in the liver (exo-erythrocytic schizogony (A), which can last a minimum of 5.5 days), the parasites undergo asexual multiplication in the erythrocytes (erythrocytic schizogony (stage B)). Merozoites leave the liver and enter the circulatory system where they infect red blood cells (5). The ring stage trophozoites mature into schizonts, which rupture releasing more merozoites, which continue the cycle of infection of RBCs (6). Some immature trophozoites, in the so-called "ring stage," differentiate into sexual erythrocytic stages (gametocytes) (7). Blood stage parasites are responsible for the clinical manifestations of the disease. 

It is important to note here (in answer to your question) that while the male and female gametocytes are formed at this stage in the human (the equivalent of meiosis), sexual reproduction does not occur until the micro and macrogametocytes are taken up by a mosquito.

The male (microgametocytes) and female (macrogametocytes) gametocytes are ingested by a female Anopheles mosquito during a blood meal (8) - only female mosquitoes (of pretty much any species) drink blood. The parasites' multiplication in the mosquito is known as the sporogonic cycle (stage C). While in the mosquito's stomach, the microgametes penetrate the macrogametes generating zygotes (9). The zygotes in turn become motile and elongated (ookinetes) (10) which invade the midgut wall of the mosquito where they develop into oocysts (11). The oocysts grow, rupture, and release sporozoites (12), which make their way to the mosquito's salivary glands. Inoculation of the sporozoites into a new human host perpetuates the malaria life cycle (1).

Sources


The Centers for Disease Control and Prevention: Malaria
Wikipedia: Plasmodium falciparum biology