Why was Sriharikota chosen by ISRO as the place to launch rockets?

Location:

Located nearer to the equator, Sriharikota is the ideal launch site for geostationary satellites.

Sriharikota is ideal for eastward launches. SHAR’s location on the east coast ensures that it gains an additional velocity of 0.4 km/s due to Earth’s rotation to easily launch rockets. Most satellites are launched eastward.  It’s near to the equator (which benefits eastward launches, rotation of the Earth provides an extra little boost, so requires less fuel). There is also a proposal for a new launch pad at Kulaseka­ra­pa­ttinam in 2013 which is ideal for southward launches. PSLV can also launch satellites into geostationary orbit (Which is above the equator with 'zero inclination' to the equatorial plane) to achieve this launch site should be ideally located on the equator or close to the equator. Otherwise it has to be maneuvered for which extra fuel is required.

Western Coastline is eliminated as mostly all the launches are eastward launches. So imagine having a launch station somewhere near Mangalore. In case of failure, our Launch Vehicle will fall on the Indian ground and can cause casualties. So we can't have a launch station on the Western Coastline. Now coming to eastern side, for the same reason as above we cant use any location till we come above Sri-lanka (Obviously we don't want our failed satellite to fall in Sri-lanka). So moving up, from Nagapattinam to Chennai, we don't have any extruding island which will be ideal for a spaceport. So the nearest one is Sriharikota.

Accessibility:

Their equipments are huge and travel from across the world. For this very reason, It must be accessible by all means of transport; Land, Air, Water. At the same time, the site is prone for accidents. So, it should be remote away from inhabitants. Being a coastal islandish area with no habitation proves advantageous for Sriharikota because it is on National Highway (NH-5), 20 KM away from nearest Railway Station, and 70 KM (Chennai) from nearest International Ports by air & ships. Therefore, most space agencies set their launch pads up near large water bodies. Baikonur (in Kazakhstan) is an exception, but it lies in the middle of an uninhabited desert, so it is safe enough. Now, in the case of India, we have two coastlines where this can be done. India has two Rocket Launch stations, the first one is at Sriharikota ,which is a barrier island off the bay of bengal coast. The other is Thumba Equatorial Rocket Launching Station in Thiruvananthapuram, very close to the southern tip of mainland India.  Now we common thing between these launching station is that they are very close to the sea and far or on the edge of the Indian Main land, this has several benefits. 

Coastal Site:

You never know what falls off from a flying object. The maximum distance of impact from the launch site can be up-to 6,500 kms. See the map towards the east of Sriharikota. So, usually coastal areas are the preferred launch sites. It also ensures that no components are shed over populated areas after the launch. All multistage parts of the rockets after the launch can be collected from the sea with minimal damage. The areas have very little population and most residents are people who work for ISRO or are local fishermen, in case of a failure there would be minimal loss of life. They have sea on one side and land on another with only one road for connectivity, which makes securing the area easier. All this makes it an ideal spaceport.

4. Climate:

Heavy rains come only in October and November allowing them to test outdoor for 10 months in a year. There's no bad day than a rain day and Sriharikota offers you optimal rainfall.

It is no accident that Sriharikota is chosen. Sriharikota is not chosen among few options. It is THE OPTION for ISRO. Sriharikota silently played & playing its role in making India proud ever since the "Rohini-125" sounding rocket was launched in 1971.