How do rare mosquito borne diseases get transmitted if they are so rare?
Via Wikipedia:

The virus is transmitted in saliva to a vertebrate host when an infected mosquito takes a blood meal. It thus cycles between mosquito and vertebrate amplifier hosts, mainly white-tailed deer. In a study from Newfoundland, JCV was significantly associated with large mammals such as sheep, cattle and horses. In Michigan and Ontario moose and bison are believed to be the primary reservoir.[3]
The virus winters in mosquito eggs, which it reaches by transovarial transmission. The female mosquito lays eggs that carry the virus, and the offspring can transmit the virus to deer or ruminants and humans. Infected mosquitoes were found equally distributed throughout the state of Connecticut, irrespective of land use.[4]

So, the virus has a reservoir in (primarily) deer, an infected deer gets bitten by a female mosquito, the mosquito lays eggs with the virus, the resulting mosquitoes can spread the disease by biting another large animal or human.
The reservoir needs to be large enough to ensure the virus survives year over year, but there doesn't need to be any special mechanism for it to travel outside the predominant range; there have been cases in several states in the northeast, though rare.

References [3] and [4] are:
[3] Goff G, Whitney H, Drebot MA (2012). &quot;Roles of host species, geographic separation, and isolation in the seroprevalence of Jamestown Canyon and snowshoe hare viruses in Newfoundland&quot;. Appl. Environ. Microbiol. 78 (18): 6734–40. doi:10.1128/AEM.01351-12. PMC 3426688. PMID 22798366.
[4] Andreadis TG, Anderson JF, Armstrong PM, Main AJ (2008). &quot;Isolations of Jamestown Canyon virus (Bunyaviridae: Orthobunyavirus) from field-collected mosquitoes (Diptera: Culicidae) in Connecticut, USA: a ten-year analysis, 1997–2006&quot;. Vector Borne Zoonotic Dis. 8 (2): 175–88. doi:10.1089/vbz.2007.0169. PMID 18386967.