What is this droplet of liquid that comes out of a mosquito?
Short answer
Excretion of blood and urine may prevent overheating by reducing body temperature through evaporative coolong (akin to perspiration). Excretion of blood and urine also concentrates the ingested blood. 

Background
Female Anopheles mosquitoes seek blood for nutrients necessary to egg production. The cold-blooded insects may excrete some freshly ingested blood as a way to avoid overheating when consuming blood as warm as 104 degrees Fahrenheit (40°C). It is thought that excretion of a droplet of fluid aids in evaporative cooling, similar to perspiration in humans, to reduce the animal's temperature. The excretion of urine may serve a similar purpose, namely avoiding overheating during blood ingestion. By keeping the urine or blood droplet attached to the abdomen, excessive heat can be eliminated (Lahonde &amp; Lazarri, 2012).

In the malaria mosquito (Anopheles stephens) the excretion of fluids has been investigated in detail. Females ingest blood meals that are equivalent to more than twice their unfed body mass. Engorged animals can fly only with great difficulty, making them more prone to predation. The blood meal serves primarily as a source of protein for egg development, but it also contains considerable amounts of unwanted salts and water that threaten haemolymph homeostasis. To counter this, a rapid natriuresis and diuresis commences even before the blood meal is completed, and ∼40% of the imbibed plasma volume and salt are voided within 1–2 h of feeding (Coast et al., 2005). 

References
- Coast et al. Exp Biol (2005); 208: 3281-91
- Lahonde &amp; Lazarri, Curr Biol (2012); 22: 40–5

Further readings
- Do mosquitoes urinate on you when they bite you?