Why do the French count so strangely?

Many languages have (at least relicts of) non-decimal counting, very often vigesimal (because we have 20 fingers plus toes), but also many other systems. I recommend an old Gutenberg project of mine, The Number Concept

Note for example that the Danish word for 55 is femoghalvtreds "five more than half the third twenty-block"

Actually, if you go back in time a bit in English, you'll realise that English was 'strange' too:

Four score and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty, and dedicated to the proposition that all men are created equal.

(The Gettysberg Address, 1863)

Now if you were to translate that into French in 1863, you'd get pretty much what you'd get today:

Il y a quatre-vingt sept ans, nos pères donnèrent naissance sur ce continent à une nouvelle nation conçue dans la liberté et vouée à la thèse selon laquelle tous les hommes sont créés égaux.

Why this has remained the case in French but has changed in English is probably more of a question for French.SE or English.SE.

actually numbers from 11 to 16 are quite regular in French (and in Italian) too: they just are a derivation from Latin.

        | French    | Italian       | Latin 
un      | on·ze     | un·dici       | un·decim
deux    | dou·ze    | do·dici       | duo·decim
trois   | trei·ze   | tre·dici      | tre·decim
quatre  | quator·ze | quattor·dici  | quattuor·decim
cinq    | quin·ze   | quin·dici     | quin·decim
six     | sei·ze    | se·dici       | se·decim
sept    | dix-sept  | dici(as)sette | septem·decim
huit    | dix-huit  | dici-otto     | duo·de·viginti
neuf    | dix-neuf  | dici(an)nove  | un·de·viginti

(18 and 19 in Latin are computed as 20-2 and 20-1). Each language has its own way to cope with small numbers.