Here you see the mouthparts of a mosquito enter a mouse's skin under a microscope. Eww!
For a start, look how flexible the mouthparts are! The tip can almost bend at right angles, and probes between the mouse’s cells in a truly sinister way. This allows the mosquito to search a large area without having to withdraw its mouthparts and start over.
“I was genuinely amazed to see the footage,” says James Logan from the London School of Hygiene and Tropical Medicine, who studies mosquitoes. “I had read that the mouthparts were mobile within the skin, but actually seeing it in real time was superb. What you assume to be a rigid structure, because it has to get into the skin like a needle, is actually flexible and fully controllable. The wonders of the insect body never cease to amaze me!”
Ed Yong has an explanation of what's going on during a mosquito bite, plus more videos, including one where the mouthparts find a blood vessel. The article might make you itch. Link