How does corona ring help against corona discharge?
Summary: 


The role of the corona ring is to distribute the electric field gradient and lower its maximum values below the corona threshold, preventing the corona discharge.*
Refer to the "superb reference" below for an extremely good explanation of theoretical and practical aspects. 




A corona ring is a conducting ring mounted on the end of or radially outside the end of an insulator in an extra high voltage system, (typically 200 kV+) 

The role of the corona ring is to eliminate or reduce ionisation of the air with associated corona discharge. Corona discharge damages insulators and may produce breakdown products which can cause catastrophic insulation failure.

A corona ring works by modifying the shape of the electric field intensity so that the worst case rate of change of field on the insulator is reduced, thereby reducing the peak potential across surrounding air to below the breakdown voltage of about 10 kV per inch (dry air, mains frequencies). 

A useful secondary role is to reshape the electric field distribution across a stack of insulators so that the potential drop per insulator is more even, thereby reducing the breakdown stresses on the insulator with highest voltage drop. A stack of symmetrical insulators used for EHV insulation will have far more voltage drop across the insulators at the "hot" end of the stack. Differences of more than 10:1 may occur in long insulator strings, with in some cases more than half the total EHV drop occurring across the first 3 or 4 insulators. The reshaping of the field by a corona ring can usefully reduce this imbalance - but a disproportionately large portion of the total voltage is still liable to be carried by the first few insulators. 



Corona rings can be seen at either end of the insulators in the photo below.



The above image is taken from this superb discussion of real world corona effects and treatment
* Thir summary of the use of corona rings is so succinct that I used it for my summary at the start of this answer. (I only found this article after I write the longer version above ;-). ) They note - The role of the corona ring is to distribute the electric field gradient and lower its maximum values below the corona threshold, preventing the corona discharge.



Wikipedia notes thatIn power systems at say 220 kV corona rings are liable to be fitted as of right to one end of insulators and to both ends at 500 kV+. 



Superb reference - Here is an extremely good paper on corona reduction and corona ring functionality - EFFECTS OF CORONA RING DESIGN ON ELECTRIC FIELD 
INTENSITY AND POTENTIAL DISTRIBUTION ALONG AN 
INSULATOR STRING

This 2010 article notes the occurrence of corona damage and insulator failure at lower voltages than has been traditionally observed, apparently due to the use of new polymer insulators. Failures and or damage have been reported on as low as 115 kV lines in some configurations, compared to expectations of not needing corona ring protection below say 161 kV in the past. 
Corona Rings: Are They Needed?
- Modeling predicts corona levels based on configuration, hardware and line voltage.

Brief but useful group discussion here

The EPRI (Electric Power research institute) publishes papers on a range of power relatd matters. Here are some of their references re use of corona rings



Illustration of electric field distributions with and without corona ring on multi insulator string. See the "superb reference" paper above for discussion.

 



Extreme examples:



From