Why do the photographs look like they do?

Or Why are these Stars Streaking?

Short answer, because of Earth's rotation.

 Long answer,
Most of these photographs have 1 Hour ( 40-90minutes) of Total exposure duration and in this much time the earth would have rotated about 15degrees.

FYI. as the earth rotates at "EXACTLY HALF" the speed of the hour hand in a regular clock,
TMI (the rotation speed of the clocks "was set to match" the rotation of the earth.)
TMI 2. If you had a watch with a 24hrs dial then it's hour hand/pointer would rotate at exactly the same speed as the earth's rotation,
the hour hand moves 90 degrees in 3 hours eg. 12 to 3 AM/PM so the earth rotates, half of 90 degrees i.e. 45degrees, so in an hour earth rotates 15 degrees, that's the Angular Length of the Streaks.

The objects on earth are stationary so they stay in proper focus while the night sky has rotated 15 degrees.

The star(s) exactly on top or below the axis (north-south) will not appear to move.
The more distant a star is from this Celestial Pole the Longer the streaks it will make.

The streaks are a good representation of the Celestial North Pole, vis a via the regular (earth's North Pole)
The height of the Pole Star above the horizon is  "your location" or Latitude with ref to the earth's equator.

In most of these photographs the Pole Star is 27* degrees above the horizon, an that's the GPS co ordinate of Jaipur 26.95 degrees or

The Circles?
When the camera is pointed towards the north, the Celestial North Pole Star or Dhruv Tara acts like a "pivot" n all the stars slowly circle it  or that's what it looks like from earth.
See the videos of these Stars in motion.

India is in the Northern Hemisphere, about 20-30 degrees north of earth's equator. This also means the Pole Star is 20-30 degrees above the Horizon.

Just bear in mind that we don't live on Top of the Earth's North Pole, so the Pole Star isn't right above your head or your Zenith.

The Grids?
The Celestial Co ordinates are "imagery lines" we make on Map of stars to basically know what part is being referred to. Like the GPS is for earth. The Celestial Longitude is divided in 24Hrs. (A very neat idea! As this is a good representation of earth's rotation speed), this is along  the North-South axis. Its tech term is RA. (Right Accession) the Zero Hrs line is though The Aries constellation.

Other info,
Colours don't matter for Astronomical photos! Yes that's a big-bad thing to say, but Astro photos are more like an X-Ray or a CAT scan than a Passport Photo.

Shooting stars is easier than seeing stars! Cameras today are more sensitive than what NASA used less than 5 years ago! 

A regular DSLR has upwards of 6400 ISO, sensitive Sensor, and in Astro Photography you remove all the Noise from the sensor/circuit/lens/dust etc. Well the cleaning or Calibrating Photos/frames is most of the work that you have to in post processing.