Why are there so many French words in English?

How many French words have entered English?
More than 33% of all English words come directly or indirectly from French. English speakers who have never studied French already know at least 10,000 French words.

Why so many?
In 1066 the Normans invaded England. They
introduced a legal and administrative system with its own vocabulary.

Around 10,000 French words came into common usage. Of these around 7,000 (judge and jury, for example) have survived into modern English.

What became known as Anglo-Norman had Latin roots. Sometimes the new Anglo-Norman words existed alongside existing Anglo-Saxon ones: beef (French) and cow (Anglo-Saxon for example.

What effect did this have on the structure of English?
It did not change the structure of the language in terms of grammar. But one very important development was that most anglo-saxon words lost their social status. The original language became grammatically simpler. Over time it evolved into what became known as Middle English.

What long-term influence did the Normans have on the language? 
Though the Norman dialect declined, French remained the language of court and learning: we still use terms like chargé d'affaires, for example.
French words became associated with learning and culture, but also with snobbishness and elitism. Fowler, in his Modern English Usage (1926) says this about the excessive use of French words and phrases :
Display of superior knowledge is as great a vulgarity as display of superior wealth — greater indeed, inasmuch as knowledge should tend more definitely than wealth towards discretion and good manners.
This combination of admiration and suspicion is still present today. Hence the joke popular in the British middle class: Pretentious? Moi?

How did French influence English pronunciation?
The introduction of French subtly modified  pronunciation in English. One example is the diphthong (long ‘o’ sound) in words like ‘boy’. Another is ‘th’ sound in  thin/shin.

The standard pronunciation of  French words these generally  approximate to the original: ‘ballet’, for example, has a silent ‘t’ rather than a sounded one as in Spanish. Some of the more common nouns have been completely anglicised - the hard ‘s’ in Paris being an obvious example.

As with many other aspects of the language, custom and practice has taken precedence over formal rules.