Ian Hiscock writes:

Theory is so important to being an independent musician.  You don't have to take your teacher's word for it that a piece should be played in a particular way; instead, you can understand chords and harmony and how that affects the direction of the melody.  You understand composition, phrasing and structure so that you get an insight into what a composer might have meant because you are also a composer yourself, working with the same issues.

Anyone working at grade 6 practical or above needs to have grade 5 theory, but there is no need to stop there, and anyone doing A level or more advanced performance should certainly study the higher theory grades.  I was fascinated to do this, and I felt really enriched by the research I did into the set works and styles for my two theory diplomas.

The written exams are a good way of testing your own progress, but they are not necessary - some people are just curious about music and how it works, and theory lessons are a way to satisfy that curiosity.

Links:

musictechteacher.com - a music theory help website set up by Karen Garrett at Green Acres Middle School, Birmingham

musictheory.net - a series of free lessons and exercises, linked to paid apps

mymusictheory.com - a full set of free support material, organised by ABRSM grades up to grade 7.  Also includes paid printable resources.

https://roselynsmusicstudio.com/theory.html - a music teacher in America with a very well-linked website - see this page for a range of theory resources.

https://www.theaterseatstore.com/blog/guide-to-music-theory - a list of the basic elements of music theory, with further links to a more detailed breakdown - and all in the blog section of a website that sells home cinema seating!  Many thanks to Danny Webb and his student Kelsey in South Carolina for sharing this!

Further reading:

Taylor, Eric: The AB Guide to Music Theory, Part I

Talyor, Eric: The AB Guide to Music Theory, Part II

Billings, Rachel: All-in-One to Grade 5 Music Theory Book (3rd edition) http://www.aaronpublications.co.uk/

Cole, Bruce: The Composer's Handbook